/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Outermost Termination of the given OTRS could not be shown: (0) OTRS (1) Thiemann-SpecialC-Transformation [EQUIVALENT, 0 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 104 ms] (4) QTRS (5) DependencyPairsProof [EQUIVALENT, 0 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) AND (9) QDP (10) UsableRulesProof [EQUIVALENT, 0 ms] (11) QDP (12) QReductionProof [EQUIVALENT, 0 ms] (13) QDP (14) UsableRulesReductionPairsProof [EQUIVALENT, 26 ms] (15) QDP (16) DependencyGraphProof [EQUIVALENT, 0 ms] (17) TRUE (18) QDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) QDP (21) QReductionProof [EQUIVALENT, 0 ms] (22) QDP (23) TransformationProof [EQUIVALENT, 0 ms] (24) QDP (25) QDPOrderProof [EQUIVALENT, 11 ms] (26) QDP (27) UsableRulesProof [EQUIVALENT, 0 ms] (28) QDP (29) QReductionProof [EQUIVALENT, 0 ms] (30) QDP (31) Trivial-Transformation [SOUND, 0 ms] (32) QTRS (33) QTRSRRRProof [EQUIVALENT, 85 ms] (34) QTRS (35) DependencyPairsProof [EQUIVALENT, 0 ms] (36) QDP (37) DependencyGraphProof [EQUIVALENT, 0 ms] (38) AND (39) QDP (40) UsableRulesProof [EQUIVALENT, 0 ms] (41) QDP (42) NonTerminationLoopProof [COMPLETE, 0 ms] (43) NO (44) QDP (45) UsableRulesProof [EQUIVALENT, 0 ms] (46) QDP (47) NonTerminationLoopProof [COMPLETE, 0 ms] (48) NO (49) Raffelsieper-Zantema-Transformation [SOUND, 0 ms] (50) QTRS (51) QTRSRRRProof [EQUIVALENT, 1024 ms] (52) QTRS (53) AAECC Innermost [EQUIVALENT, 773 ms] (54) QTRS (55) DependencyPairsProof [EQUIVALENT, 213 ms] (56) QDP (57) DependencyGraphProof [EQUIVALENT, 0 ms] (58) AND (59) QDP (60) UsableRulesProof [EQUIVALENT, 0 ms] (61) QDP (62) QReductionProof [EQUIVALENT, 0 ms] (63) QDP (64) QDPSizeChangeProof [EQUIVALENT, 0 ms] (65) YES (66) QDP (67) UsableRulesProof [EQUIVALENT, 0 ms] (68) QDP (69) QReductionProof [EQUIVALENT, 0 ms] (70) QDP (71) TransformationProof [EQUIVALENT, 193 ms] (72) QDP (73) DependencyGraphProof [EQUIVALENT, 0 ms] (74) QDP (75) TransformationProof [EQUIVALENT, 2 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) DependencyGraphProof [EQUIVALENT, 0 ms] (96) QDP (97) TransformationProof [EQUIVALENT, 0 ms] (98) QDP (99) DependencyGraphProof [EQUIVALENT, 0 ms] (100) QDP (101) TransformationProof [EQUIVALENT, 4 ms] (102) QDP (103) TransformationProof [EQUIVALENT, 0 ms] (104) QDP (105) DependencyGraphProof [EQUIVALENT, 0 ms] (106) QDP (107) TransformationProof [EQUIVALENT, 0 ms] (108) QDP (109) DependencyGraphProof [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, 4 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, 0 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, 2 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, 3 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, 0 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, 1 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) DependencyGraphProof [EQUIVALENT, 0 ms] (276) QDP (277) TransformationProof [EQUIVALENT, 0 ms] (278) QDP (279) DependencyGraphProof [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, 1 ms] (292) QDP (293) TransformationProof [EQUIVALENT, 0 ms] (294) QDP (295) DependencyGraphProof [EQUIVALENT, 0 ms] (296) QDP (297) TransformationProof [EQUIVALENT, 0 ms] (298) QDP (299) DependencyGraphProof [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, 0 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, 2 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, 3 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) DependencyGraphProof [EQUIVALENT, 0 ms] (444) QDP (445) TransformationProof [EQUIVALENT, 1 ms] (446) QDP (447) TransformationProof [EQUIVALENT, 0 ms] (448) QDP (449) TransformationProof [EQUIVALENT, 0 ms] (450) QDP (451) DependencyGraphProof [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) DependencyGraphProof [EQUIVALENT, 0 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) DependencyGraphProof [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, 5 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, 1 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, 3 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, 0 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) DependencyGraphProof [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) DependencyGraphProof [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) DependencyGraphProof [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) DependencyGraphProof [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, 0 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, 1 ms] (740) QDP (741) TransformationProof [EQUIVALENT, 0 ms] (742) QDP (743) DependencyGraphProof [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) DependencyGraphProof [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) DependencyGraphProof [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) DependencyGraphProof [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, 2 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, 0 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) DependencyGraphProof [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) DependencyGraphProof [EQUIVALENT, 0 ms] (880) QDP (881) TransformationProof [EQUIVALENT, 3 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) DependencyGraphProof [EQUIVALENT, 0 ms] (896) QDP (897) TransformationProof [EQUIVALENT, 0 ms] (898) QDP (899) TransformationProof [EQUIVALENT, 0 ms] (900) QDP (901) TransformationProof [EQUIVALENT, 0 ms] (902) QDP (903) DependencyGraphProof [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, 0 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) DependencyGraphProof [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) DependencyGraphProof [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) DependencyGraphProof [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) DependencyGraphProof [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, 0 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) DependencyGraphProof [EQUIVALENT, 0 ms] (1072) QDP (1073) TransformationProof [EQUIVALENT, 0 ms] (1074) QDP (1075) TransformationProof [EQUIVALENT, 0 ms] (1076) QDP (1077) TransformationProof [EQUIVALENT, 0 ms] (1078) QDP (1079) DependencyGraphProof [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) DependencyGraphProof [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) DependencyGraphProof [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, 0 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, 1 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) TransformationProof [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, 0 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) TransformationProof [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) DependencyGraphProof [EQUIVALENT, 0 ms] (1284) QDP (1285) TransformationProof [EQUIVALENT, 0 ms] (1286) QDP (1287) TransformationProof [EQUIVALENT, 0 ms] (1288) QDP (1289) TransformationProof [EQUIVALENT, 0 ms] (1290) QDP (1291) TransformationProof [EQUIVALENT, 0 ms] (1292) QDP (1293) DependencyGraphProof [EQUIVALENT, 0 ms] (1294) QDP (1295) QDPOrderProof [EQUIVALENT, 72 ms] (1296) QDP (1297) QDPOrderProof [EQUIVALENT, 59 ms] (1298) QDP (1299) MNOCProof [EQUIVALENT, 0 ms] (1300) QDP (1301) SplitQDPProof [EQUIVALENT, 0 ms] (1302) AND (1303) QDP (1304) SemLabProof [SOUND, 0 ms] (1305) QDP (1306) UsableRulesReductionPairsProof [EQUIVALENT, 130 ms] (1307) QDP (1308) MRRProof [EQUIVALENT, 141 ms] (1309) QDP (1310) DependencyGraphProof [EQUIVALENT, 0 ms] (1311) QDP (1312) PisEmptyProof [SOUND, 0 ms] (1313) TRUE (1314) QDP (1315) SplitQDPProof [EQUIVALENT, 0 ms] (1316) AND (1317) QDP (1318) SemLabProof [SOUND, 0 ms] (1319) QDP (1320) UsableRulesReductionPairsProof [EQUIVALENT, 172 ms] (1321) QDP (1322) MRRProof [EQUIVALENT, 70 ms] (1323) QDP (1324) DependencyGraphProof [EQUIVALENT, 0 ms] (1325) QDP (1326) QDPOrderProof [EQUIVALENT, 51 ms] (1327) QDP (1328) PisEmptyProof [SOUND, 0 ms] (1329) TRUE (1330) QDP (1331) SplitQDPProof [EQUIVALENT, 0 ms] (1332) AND (1333) QDP (1334) SemLabProof [SOUND, 0 ms] (1335) QDP (1336) UsableRulesReductionPairsProof [EQUIVALENT, 159 ms] (1337) QDP (1338) MRRProof [EQUIVALENT, 69 ms] (1339) QDP (1340) DependencyGraphProof [EQUIVALENT, 0 ms] (1341) QDP (1342) QDPOrderProof [EQUIVALENT, 58 ms] (1343) QDP (1344) PisEmptyProof [SOUND, 0 ms] (1345) TRUE (1346) QDP (1347) SplitQDPProof [EQUIVALENT, 0 ms] (1348) AND (1349) QDP (1350) SemLabProof [SOUND, 0 ms] (1351) QDP (1352) MRRProof [EQUIVALENT, 105 ms] (1353) QDP (1354) DependencyGraphProof [EQUIVALENT, 0 ms] (1355) QDP (1356) PisEmptyProof [SOUND, 0 ms] (1357) TRUE (1358) QDP (1359) SplitQDPProof [EQUIVALENT, 0 ms] (1360) AND (1361) QDP (1362) SemLabProof [SOUND, 0 ms] (1363) QDP (1364) MRRProof [EQUIVALENT, 133 ms] (1365) QDP (1366) DependencyGraphProof [EQUIVALENT, 0 ms] (1367) QDP (1368) UsableRulesReductionPairsProof [EQUIVALENT, 81 ms] (1369) QDP (1370) MRRProof [EQUIVALENT, 47 ms] (1371) QDP (1372) PisEmptyProof [SOUND, 0 ms] (1373) TRUE (1374) QDP ---------------------------------------- (0) Obligation: Term rewrite system R: The TRS R consists of the following rules: a -> f(a) b -> g(b) a -> b f(f(f(f(f(f(f(f(f(f(x)))))))))) -> c g(g(g(g(g(g(g(g(g(g(x)))))))))) -> c Outermost Strategy. ---------------------------------------- (1) Thiemann-SpecialC-Transformation (EQUIVALENT) We applied the Thiemann-SpecialC transformation to transform the outermost TRS to an innermost TRS. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(a) -> go_up(f(a)) reduce(b) -> go_up(g(b)) reduce(a) -> go_up(b) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) check_f(result_f(x)) -> go_up(x) check_g(result_g(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(g(x0)) reduce(a) reduce(b) redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0)) check_g(redex_g(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a) = 2 POL(b) = 1 POL(c) = 0 POL(check_f(x_1)) = 2*x_1 POL(check_g(x_1)) = 2*x_1 POL(f(x_1)) = x_1 POL(g(x_1)) = x_1 POL(go_up(x_1)) = 2*x_1 POL(in_f_1(x_1)) = x_1 POL(in_g_1(x_1)) = x_1 POL(redex_f(x_1)) = x_1 POL(redex_g(x_1)) = x_1 POL(reduce(x_1)) = 2*x_1 POL(result_f(x_1)) = 2*x_1 POL(result_g(x_1)) = x_1 POL(top(x_1)) = 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: reduce(a) -> go_up(b) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(a) -> go_up(f(a)) reduce(b) -> go_up(g(b)) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) check_f(result_f(x)) -> go_up(x) check_g(result_g(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(g(x0)) reduce(a) reduce(b) redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0)) check_g(redex_g(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) ---------------------------------------- (5) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) TOP(go_up(x)) -> REDUCE(x) REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) REDUCE(f(x_1)) -> REDEX_F(x_1) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) REDUCE(g(x_1)) -> REDEX_G(x_1) CHECK_F(redex_f(x_1)) -> IN_F_1(reduce(x_1)) CHECK_F(redex_f(x_1)) -> REDUCE(x_1) CHECK_G(redex_g(x_1)) -> IN_G_1(reduce(x_1)) CHECK_G(redex_g(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(a) -> go_up(f(a)) reduce(b) -> go_up(g(b)) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) check_f(result_f(x)) -> go_up(x) check_g(result_g(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(g(x0)) reduce(a) reduce(b) redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0)) check_g(redex_g(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes. ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_F(redex_f(x_1)) -> REDUCE(x_1) REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) CHECK_G(redex_g(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(a) -> go_up(f(a)) reduce(b) -> go_up(g(b)) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) check_f(result_f(x)) -> go_up(x) check_g(result_g(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(g(x0)) reduce(a) reduce(b) redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0)) check_g(redex_g(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_F(redex_f(x_1)) -> REDUCE(x_1) REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) CHECK_G(redex_g(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(g(x0)) reduce(a) reduce(b) redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0)) check_g(redex_g(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(go_up(x0)) reduce(f(x0)) reduce(g(x0)) reduce(a) reduce(b) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0)) check_g(redex_g(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_F(redex_f(x_1)) -> REDUCE(x_1) REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) CHECK_G(redex_g(x_1)) -> REDUCE(x_1) The TRS R consists of the following rules: redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) The set Q consists of the following terms: redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) 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. The following dependency pairs can be deleted: REDUCE(f(x_1)) -> CHECK_F(redex_f(x_1)) REDUCE(g(x_1)) -> CHECK_G(redex_g(x_1)) The following rules are removed from R: redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) Used ordering: POLO with Polynomial interpretation [POLO]: POL(CHECK_F(x_1)) = x_1 POL(CHECK_G(x_1)) = x_1 POL(REDUCE(x_1)) = 2*x_1 POL(c) = 0 POL(f(x_1)) = 2*x_1 POL(g(x_1)) = 2*x_1 POL(redex_f(x_1)) = 2*x_1 POL(redex_g(x_1)) = 2*x_1 POL(result_f(x_1)) = x_1 POL(result_g(x_1)) = x_1 ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: CHECK_F(redex_f(x_1)) -> REDUCE(x_1) CHECK_G(redex_g(x_1)) -> REDUCE(x_1) R is empty. The set Q consists of the following terms: redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (17) TRUE ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: top(go_up(x)) -> top(reduce(x)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(a) -> go_up(f(a)) reduce(b) -> go_up(g(b)) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) check_f(result_f(x)) -> go_up(x) check_g(result_g(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(g(x0)) reduce(a) reduce(b) redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0)) check_g(redex_g(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(a) -> go_up(f(a)) reduce(b) -> go_up(g(b)) redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(g(x0)) reduce(a) reduce(b) redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0)) check_g(redex_g(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(go_up(x0)) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(a) -> go_up(f(a)) reduce(b) -> go_up(g(b)) redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(g(x0)) reduce(a) reduce(b) redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0)) check_g(redex_g(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule TOP(go_up(x)) -> TOP(reduce(x)) at position [0] we obtained the following new rules [LPAR04]: (TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))),TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0)))) (TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))),TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0)))) (TOP(go_up(a)) -> TOP(go_up(f(a))),TOP(go_up(a)) -> TOP(go_up(f(a)))) (TOP(go_up(b)) -> TOP(go_up(g(b))),TOP(go_up(b)) -> TOP(go_up(g(b)))) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))) TOP(go_up(a)) -> TOP(go_up(f(a))) TOP(go_up(b)) -> TOP(go_up(g(b))) The TRS R consists of the following rules: reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(a) -> go_up(f(a)) reduce(b) -> go_up(g(b)) redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(g(x0)) reduce(a) reduce(b) redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0)) check_g(redex_g(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP(go_up(a)) -> TOP(go_up(f(a))) TOP(go_up(b)) -> TOP(go_up(g(b))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(TOP(x_1)) = x_1 POL(a) = 1 POL(b) = 1 POL(c) = 0 POL(check_f(x_1)) = x_1 POL(check_g(x_1)) = x_1 POL(f(x_1)) = 0 POL(g(x_1)) = 0 POL(go_up(x_1)) = x_1 POL(in_f_1(x_1)) = 0 POL(in_g_1(x_1)) = 0 POL(redex_f(x_1)) = 0 POL(redex_g(x_1)) = 0 POL(reduce(x_1)) = 0 POL(result_f(x_1)) = x_1 POL(result_g(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(g(x_1)) -> check_g(redex_g(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(f(x0))) -> TOP(check_f(redex_f(x0))) TOP(go_up(g(x0))) -> TOP(check_g(redex_g(x0))) The TRS R consists of the following rules: reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(a) -> go_up(f(a)) reduce(b) -> go_up(g(b)) redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(g(x0)) reduce(a) reduce(b) redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0)) check_g(redex_g(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(a) -> go_up(f(a)) reduce(b) -> go_up(g(b)) redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: top(go_up(x0)) reduce(f(x0)) reduce(g(x0)) reduce(a) reduce(b) redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0)) check_g(redex_g(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(go_up(x0)) ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(go_up(x)) -> TOP(reduce(x)) The TRS R consists of the following rules: reduce(f(x_1)) -> check_f(redex_f(x_1)) reduce(g(x_1)) -> check_g(redex_g(x_1)) reduce(a) -> go_up(f(a)) reduce(b) -> go_up(g(b)) redex_g(g(g(g(g(g(g(g(g(g(x)))))))))) -> result_g(c) check_g(result_g(x)) -> go_up(x) check_g(redex_g(x_1)) -> in_g_1(reduce(x_1)) in_g_1(go_up(x_1)) -> go_up(g(x_1)) redex_f(f(f(f(f(f(f(f(f(f(x)))))))))) -> result_f(c) check_f(result_f(x)) -> go_up(x) check_f(redex_f(x_1)) -> in_f_1(reduce(x_1)) in_f_1(go_up(x_1)) -> go_up(f(x_1)) The set Q consists of the following terms: reduce(f(x0)) reduce(g(x0)) reduce(a) reduce(b) redex_f(f(f(f(f(f(f(f(f(f(x0)))))))))) redex_g(g(g(g(g(g(g(g(g(g(x0)))))))))) check_f(result_f(x0)) check_g(result_g(x0)) check_f(redex_f(x0)) check_g(redex_g(x0)) in_f_1(go_up(x0)) in_g_1(go_up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) Trivial-Transformation (SOUND) We applied the Trivial transformation to transform the outermost TRS to a standard TRS. ---------------------------------------- (32) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a -> f(a) b -> g(b) a -> b f(f(f(f(f(f(f(f(f(f(x)))))))))) -> c g(g(g(g(g(g(g(g(g(g(x)))))))))) -> c Q is empty. ---------------------------------------- (33) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a) = 1 POL(b) = 0 POL(c) = 0 POL(f(x_1)) = x_1 POL(g(x_1)) = 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a -> b ---------------------------------------- (34) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a -> f(a) b -> g(b) f(f(f(f(f(f(f(f(f(f(x)))))))))) -> c g(g(g(g(g(g(g(g(g(g(x)))))))))) -> c Q is empty. ---------------------------------------- (35) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: A -> F(a) A -> A B -> G(b) B -> B The TRS R consists of the following rules: a -> f(a) b -> g(b) f(f(f(f(f(f(f(f(f(f(x)))))))))) -> c g(g(g(g(g(g(g(g(g(g(x)))))))))) -> c Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. ---------------------------------------- (38) Complex Obligation (AND) ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: B -> B The TRS R consists of the following rules: a -> f(a) b -> g(b) f(f(f(f(f(f(f(f(f(f(x)))))))))) -> c g(g(g(g(g(g(g(g(g(g(x)))))))))) -> c Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (40) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (41) Obligation: Q DP problem: The TRS P consists of the following rules: B -> B R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (42) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = B evaluates to t =B Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from B to B. ---------------------------------------- (43) NO ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: A -> A The TRS R consists of the following rules: a -> f(a) b -> g(b) f(f(f(f(f(f(f(f(f(f(x)))))))))) -> c g(g(g(g(g(g(g(g(g(g(x)))))))))) -> c Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: A -> A R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = A evaluates to t =A Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from A to A. ---------------------------------------- (48) NO ---------------------------------------- (49) Raffelsieper-Zantema-Transformation (SOUND) We applied the Raffelsieper-Zantema transformation to transform the outermost TRS to a standard TRS. ---------------------------------------- (50) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(a) -> up(b) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) Q is empty. ---------------------------------------- (51) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a) = 1 POL(b) = 0 POL(c) = 0 POL(down(x_1)) = 2*x_1 POL(f(x_1)) = x_1 POL(f_flat(x_1)) = x_1 POL(fresh_constant) = 0 POL(g(x_1)) = x_1 POL(g_flat(x_1)) = x_1 POL(top(x_1)) = 2*x_1 POL(up(x_1)) = 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: down(a) -> up(b) ---------------------------------------- (52) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) Q is empty. ---------------------------------------- (53) AAECC Innermost (EQUIVALENT) We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) The TRS R 2 is top(up(x)) -> top(down(x)) The signature Sigma is {top_1} ---------------------------------------- (54) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) top(up(x0)) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) ---------------------------------------- (55) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) TOP(up(x)) -> DOWN(x) DOWN(f(a)) -> F_FLAT(down(a)) DOWN(f(a)) -> DOWN(a) DOWN(f(b)) -> F_FLAT(down(b)) DOWN(f(b)) -> DOWN(b) DOWN(f(g(y4))) -> F_FLAT(down(g(y4))) DOWN(f(g(y4))) -> DOWN(g(y4)) DOWN(f(c)) -> F_FLAT(down(c)) DOWN(f(c)) -> DOWN(c) DOWN(f(fresh_constant)) -> F_FLAT(down(fresh_constant)) DOWN(f(fresh_constant)) -> DOWN(fresh_constant) DOWN(g(a)) -> G_FLAT(down(a)) DOWN(g(a)) -> DOWN(a) DOWN(g(f(y6))) -> G_FLAT(down(f(y6))) DOWN(g(f(y6))) -> DOWN(f(y6)) DOWN(g(b)) -> G_FLAT(down(b)) DOWN(g(b)) -> DOWN(b) DOWN(g(c)) -> G_FLAT(down(c)) DOWN(g(c)) -> DOWN(c) DOWN(g(fresh_constant)) -> G_FLAT(down(fresh_constant)) DOWN(g(fresh_constant)) -> DOWN(fresh_constant) DOWN(f(f(a))) -> F_FLAT(down(f(a))) DOWN(f(f(a))) -> DOWN(f(a)) DOWN(f(f(b))) -> F_FLAT(down(f(b))) DOWN(f(f(b))) -> DOWN(f(b)) DOWN(f(f(g(y10)))) -> F_FLAT(down(f(g(y10)))) DOWN(f(f(g(y10)))) -> DOWN(f(g(y10))) DOWN(f(f(c))) -> F_FLAT(down(f(c))) DOWN(f(f(c))) -> DOWN(f(c)) DOWN(f(f(fresh_constant))) -> F_FLAT(down(f(fresh_constant))) DOWN(f(f(fresh_constant))) -> DOWN(f(fresh_constant)) DOWN(g(g(a))) -> G_FLAT(down(g(a))) DOWN(g(g(a))) -> DOWN(g(a)) DOWN(g(g(f(y12)))) -> G_FLAT(down(g(f(y12)))) DOWN(g(g(f(y12)))) -> DOWN(g(f(y12))) DOWN(g(g(b))) -> G_FLAT(down(g(b))) DOWN(g(g(b))) -> DOWN(g(b)) DOWN(g(g(c))) -> G_FLAT(down(g(c))) DOWN(g(g(c))) -> DOWN(g(c)) DOWN(g(g(fresh_constant))) -> G_FLAT(down(g(fresh_constant))) DOWN(g(g(fresh_constant))) -> DOWN(g(fresh_constant)) DOWN(f(f(f(a)))) -> F_FLAT(down(f(f(a)))) DOWN(f(f(f(a)))) -> DOWN(f(f(a))) DOWN(f(f(f(b)))) -> F_FLAT(down(f(f(b)))) DOWN(f(f(f(b)))) -> DOWN(f(f(b))) DOWN(f(f(f(g(y16))))) -> F_FLAT(down(f(f(g(y16))))) DOWN(f(f(f(g(y16))))) -> DOWN(f(f(g(y16)))) DOWN(f(f(f(c)))) -> F_FLAT(down(f(f(c)))) DOWN(f(f(f(c)))) -> DOWN(f(f(c))) DOWN(f(f(f(fresh_constant)))) -> F_FLAT(down(f(f(fresh_constant)))) DOWN(f(f(f(fresh_constant)))) -> DOWN(f(f(fresh_constant))) DOWN(g(g(g(a)))) -> G_FLAT(down(g(g(a)))) DOWN(g(g(g(a)))) -> DOWN(g(g(a))) DOWN(g(g(g(f(y18))))) -> G_FLAT(down(g(g(f(y18))))) DOWN(g(g(g(f(y18))))) -> DOWN(g(g(f(y18)))) DOWN(g(g(g(b)))) -> G_FLAT(down(g(g(b)))) DOWN(g(g(g(b)))) -> DOWN(g(g(b))) DOWN(g(g(g(c)))) -> G_FLAT(down(g(g(c)))) DOWN(g(g(g(c)))) -> DOWN(g(g(c))) DOWN(g(g(g(fresh_constant)))) -> G_FLAT(down(g(g(fresh_constant)))) DOWN(g(g(g(fresh_constant)))) -> DOWN(g(g(fresh_constant))) DOWN(f(f(f(f(a))))) -> F_FLAT(down(f(f(f(a))))) DOWN(f(f(f(f(a))))) -> DOWN(f(f(f(a)))) DOWN(f(f(f(f(b))))) -> F_FLAT(down(f(f(f(b))))) DOWN(f(f(f(f(b))))) -> DOWN(f(f(f(b)))) DOWN(f(f(f(f(g(y22)))))) -> F_FLAT(down(f(f(f(g(y22)))))) DOWN(f(f(f(f(g(y22)))))) -> DOWN(f(f(f(g(y22))))) DOWN(f(f(f(f(c))))) -> F_FLAT(down(f(f(f(c))))) DOWN(f(f(f(f(c))))) -> DOWN(f(f(f(c)))) DOWN(f(f(f(f(fresh_constant))))) -> F_FLAT(down(f(f(f(fresh_constant))))) DOWN(f(f(f(f(fresh_constant))))) -> DOWN(f(f(f(fresh_constant)))) DOWN(g(g(g(g(a))))) -> G_FLAT(down(g(g(g(a))))) DOWN(g(g(g(g(a))))) -> DOWN(g(g(g(a)))) DOWN(g(g(g(g(f(y24)))))) -> G_FLAT(down(g(g(g(f(y24)))))) DOWN(g(g(g(g(f(y24)))))) -> DOWN(g(g(g(f(y24))))) DOWN(g(g(g(g(b))))) -> G_FLAT(down(g(g(g(b))))) DOWN(g(g(g(g(b))))) -> DOWN(g(g(g(b)))) DOWN(g(g(g(g(c))))) -> G_FLAT(down(g(g(g(c))))) DOWN(g(g(g(g(c))))) -> DOWN(g(g(g(c)))) DOWN(g(g(g(g(fresh_constant))))) -> G_FLAT(down(g(g(g(fresh_constant))))) DOWN(g(g(g(g(fresh_constant))))) -> DOWN(g(g(g(fresh_constant)))) DOWN(f(f(f(f(f(a)))))) -> F_FLAT(down(f(f(f(f(a)))))) DOWN(f(f(f(f(f(a)))))) -> DOWN(f(f(f(f(a))))) DOWN(f(f(f(f(f(b)))))) -> F_FLAT(down(f(f(f(f(b)))))) DOWN(f(f(f(f(f(b)))))) -> DOWN(f(f(f(f(b))))) DOWN(f(f(f(f(f(g(y28))))))) -> F_FLAT(down(f(f(f(f(g(y28))))))) DOWN(f(f(f(f(f(g(y28))))))) -> DOWN(f(f(f(f(g(y28)))))) DOWN(f(f(f(f(f(c)))))) -> F_FLAT(down(f(f(f(f(c)))))) DOWN(f(f(f(f(f(c)))))) -> DOWN(f(f(f(f(c))))) DOWN(f(f(f(f(f(fresh_constant)))))) -> F_FLAT(down(f(f(f(f(fresh_constant)))))) DOWN(f(f(f(f(f(fresh_constant)))))) -> DOWN(f(f(f(f(fresh_constant))))) DOWN(g(g(g(g(g(a)))))) -> G_FLAT(down(g(g(g(g(a)))))) DOWN(g(g(g(g(g(a)))))) -> DOWN(g(g(g(g(a))))) DOWN(g(g(g(g(g(f(y30))))))) -> G_FLAT(down(g(g(g(g(f(y30))))))) DOWN(g(g(g(g(g(f(y30))))))) -> DOWN(g(g(g(g(f(y30)))))) DOWN(g(g(g(g(g(b)))))) -> G_FLAT(down(g(g(g(g(b)))))) DOWN(g(g(g(g(g(b)))))) -> DOWN(g(g(g(g(b))))) DOWN(g(g(g(g(g(c)))))) -> G_FLAT(down(g(g(g(g(c)))))) DOWN(g(g(g(g(g(c)))))) -> DOWN(g(g(g(g(c))))) DOWN(g(g(g(g(g(fresh_constant)))))) -> G_FLAT(down(g(g(g(g(fresh_constant)))))) DOWN(g(g(g(g(g(fresh_constant)))))) -> DOWN(g(g(g(g(fresh_constant))))) DOWN(f(f(f(f(f(f(a))))))) -> F_FLAT(down(f(f(f(f(f(a))))))) DOWN(f(f(f(f(f(f(a))))))) -> DOWN(f(f(f(f(f(a)))))) DOWN(f(f(f(f(f(f(b))))))) -> F_FLAT(down(f(f(f(f(f(b))))))) DOWN(f(f(f(f(f(f(b))))))) -> DOWN(f(f(f(f(f(b)))))) DOWN(f(f(f(f(f(f(g(y34)))))))) -> F_FLAT(down(f(f(f(f(f(g(y34)))))))) DOWN(f(f(f(f(f(f(g(y34)))))))) -> DOWN(f(f(f(f(f(g(y34))))))) DOWN(f(f(f(f(f(f(c))))))) -> F_FLAT(down(f(f(f(f(f(c))))))) DOWN(f(f(f(f(f(f(c))))))) -> DOWN(f(f(f(f(f(c)))))) DOWN(f(f(f(f(f(f(fresh_constant))))))) -> F_FLAT(down(f(f(f(f(f(fresh_constant))))))) DOWN(f(f(f(f(f(f(fresh_constant))))))) -> DOWN(f(f(f(f(f(fresh_constant)))))) DOWN(g(g(g(g(g(g(a))))))) -> G_FLAT(down(g(g(g(g(g(a))))))) DOWN(g(g(g(g(g(g(a))))))) -> DOWN(g(g(g(g(g(a)))))) DOWN(g(g(g(g(g(g(f(y36)))))))) -> G_FLAT(down(g(g(g(g(g(f(y36)))))))) DOWN(g(g(g(g(g(g(f(y36)))))))) -> DOWN(g(g(g(g(g(f(y36))))))) DOWN(g(g(g(g(g(g(b))))))) -> G_FLAT(down(g(g(g(g(g(b))))))) DOWN(g(g(g(g(g(g(b))))))) -> DOWN(g(g(g(g(g(b)))))) DOWN(g(g(g(g(g(g(c))))))) -> G_FLAT(down(g(g(g(g(g(c))))))) DOWN(g(g(g(g(g(g(c))))))) -> DOWN(g(g(g(g(g(c)))))) DOWN(g(g(g(g(g(g(fresh_constant))))))) -> G_FLAT(down(g(g(g(g(g(fresh_constant))))))) DOWN(g(g(g(g(g(g(fresh_constant))))))) -> DOWN(g(g(g(g(g(fresh_constant)))))) DOWN(f(f(f(f(f(f(f(a)))))))) -> F_FLAT(down(f(f(f(f(f(f(a)))))))) DOWN(f(f(f(f(f(f(f(a)))))))) -> DOWN(f(f(f(f(f(f(a))))))) DOWN(f(f(f(f(f(f(f(b)))))))) -> F_FLAT(down(f(f(f(f(f(f(b)))))))) DOWN(f(f(f(f(f(f(f(b)))))))) -> DOWN(f(f(f(f(f(f(b))))))) DOWN(f(f(f(f(f(f(f(g(y40))))))))) -> F_FLAT(down(f(f(f(f(f(f(g(y40))))))))) DOWN(f(f(f(f(f(f(f(g(y40))))))))) -> DOWN(f(f(f(f(f(f(g(y40)))))))) DOWN(f(f(f(f(f(f(f(c)))))))) -> F_FLAT(down(f(f(f(f(f(f(c)))))))) DOWN(f(f(f(f(f(f(f(c)))))))) -> DOWN(f(f(f(f(f(f(c))))))) DOWN(f(f(f(f(f(f(f(fresh_constant)))))))) -> F_FLAT(down(f(f(f(f(f(f(fresh_constant)))))))) DOWN(f(f(f(f(f(f(f(fresh_constant)))))))) -> DOWN(f(f(f(f(f(f(fresh_constant))))))) DOWN(g(g(g(g(g(g(g(a)))))))) -> G_FLAT(down(g(g(g(g(g(g(a)))))))) DOWN(g(g(g(g(g(g(g(a)))))))) -> DOWN(g(g(g(g(g(g(a))))))) DOWN(g(g(g(g(g(g(g(f(y42))))))))) -> G_FLAT(down(g(g(g(g(g(g(f(y42))))))))) DOWN(g(g(g(g(g(g(g(f(y42))))))))) -> DOWN(g(g(g(g(g(g(f(y42)))))))) DOWN(g(g(g(g(g(g(g(b)))))))) -> G_FLAT(down(g(g(g(g(g(g(b)))))))) DOWN(g(g(g(g(g(g(g(b)))))))) -> DOWN(g(g(g(g(g(g(b))))))) DOWN(g(g(g(g(g(g(g(c)))))))) -> G_FLAT(down(g(g(g(g(g(g(c)))))))) DOWN(g(g(g(g(g(g(g(c)))))))) -> DOWN(g(g(g(g(g(g(c))))))) DOWN(g(g(g(g(g(g(g(fresh_constant)))))))) -> G_FLAT(down(g(g(g(g(g(g(fresh_constant)))))))) DOWN(g(g(g(g(g(g(g(fresh_constant)))))))) -> DOWN(g(g(g(g(g(g(fresh_constant))))))) DOWN(f(f(f(f(f(f(f(f(a))))))))) -> F_FLAT(down(f(f(f(f(f(f(f(a))))))))) DOWN(f(f(f(f(f(f(f(f(a))))))))) -> DOWN(f(f(f(f(f(f(f(a)))))))) DOWN(f(f(f(f(f(f(f(f(b))))))))) -> F_FLAT(down(f(f(f(f(f(f(f(b))))))))) DOWN(f(f(f(f(f(f(f(f(b))))))))) -> DOWN(f(f(f(f(f(f(f(b)))))))) DOWN(f(f(f(f(f(f(f(f(g(y46)))))))))) -> F_FLAT(down(f(f(f(f(f(f(f(g(y46)))))))))) DOWN(f(f(f(f(f(f(f(f(g(y46)))))))))) -> DOWN(f(f(f(f(f(f(f(g(y46))))))))) DOWN(f(f(f(f(f(f(f(f(c))))))))) -> F_FLAT(down(f(f(f(f(f(f(f(c))))))))) DOWN(f(f(f(f(f(f(f(f(c))))))))) -> DOWN(f(f(f(f(f(f(f(c)))))))) DOWN(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> F_FLAT(down(f(f(f(f(f(f(f(fresh_constant))))))))) DOWN(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> DOWN(f(f(f(f(f(f(f(fresh_constant)))))))) DOWN(g(g(g(g(g(g(g(g(a))))))))) -> G_FLAT(down(g(g(g(g(g(g(g(a))))))))) DOWN(g(g(g(g(g(g(g(g(a))))))))) -> DOWN(g(g(g(g(g(g(g(a)))))))) DOWN(g(g(g(g(g(g(g(g(f(y48)))))))))) -> G_FLAT(down(g(g(g(g(g(g(g(f(y48)))))))))) DOWN(g(g(g(g(g(g(g(g(f(y48)))))))))) -> DOWN(g(g(g(g(g(g(g(f(y48))))))))) DOWN(g(g(g(g(g(g(g(g(b))))))))) -> G_FLAT(down(g(g(g(g(g(g(g(b))))))))) DOWN(g(g(g(g(g(g(g(g(b))))))))) -> DOWN(g(g(g(g(g(g(g(b)))))))) DOWN(g(g(g(g(g(g(g(g(c))))))))) -> G_FLAT(down(g(g(g(g(g(g(g(c))))))))) DOWN(g(g(g(g(g(g(g(g(c))))))))) -> DOWN(g(g(g(g(g(g(g(c)))))))) DOWN(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> G_FLAT(down(g(g(g(g(g(g(g(fresh_constant))))))))) DOWN(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> DOWN(g(g(g(g(g(g(g(fresh_constant)))))))) DOWN(f(f(f(f(f(f(f(f(f(a)))))))))) -> F_FLAT(down(f(f(f(f(f(f(f(f(a)))))))))) DOWN(f(f(f(f(f(f(f(f(f(a)))))))))) -> DOWN(f(f(f(f(f(f(f(f(a))))))))) DOWN(f(f(f(f(f(f(f(f(f(b)))))))))) -> F_FLAT(down(f(f(f(f(f(f(f(f(b)))))))))) DOWN(f(f(f(f(f(f(f(f(f(b)))))))))) -> DOWN(f(f(f(f(f(f(f(f(b))))))))) DOWN(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> F_FLAT(down(f(f(f(f(f(f(f(f(g(y52))))))))))) DOWN(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> DOWN(f(f(f(f(f(f(f(f(g(y52)))))))))) DOWN(f(f(f(f(f(f(f(f(f(c)))))))))) -> F_FLAT(down(f(f(f(f(f(f(f(f(c)))))))))) DOWN(f(f(f(f(f(f(f(f(f(c)))))))))) -> DOWN(f(f(f(f(f(f(f(f(c))))))))) DOWN(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> F_FLAT(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) DOWN(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> DOWN(f(f(f(f(f(f(f(f(fresh_constant))))))))) DOWN(g(g(g(g(g(g(g(g(g(a)))))))))) -> G_FLAT(down(g(g(g(g(g(g(g(g(a)))))))))) DOWN(g(g(g(g(g(g(g(g(g(a)))))))))) -> DOWN(g(g(g(g(g(g(g(g(a))))))))) DOWN(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> G_FLAT(down(g(g(g(g(g(g(g(g(f(y54))))))))))) DOWN(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> DOWN(g(g(g(g(g(g(g(g(f(y54)))))))))) DOWN(g(g(g(g(g(g(g(g(g(b)))))))))) -> G_FLAT(down(g(g(g(g(g(g(g(g(b)))))))))) DOWN(g(g(g(g(g(g(g(g(g(b)))))))))) -> DOWN(g(g(g(g(g(g(g(g(b))))))))) DOWN(g(g(g(g(g(g(g(g(g(c)))))))))) -> G_FLAT(down(g(g(g(g(g(g(g(g(c)))))))))) DOWN(g(g(g(g(g(g(g(g(g(c)))))))))) -> DOWN(g(g(g(g(g(g(g(g(c))))))))) DOWN(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> G_FLAT(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) DOWN(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> DOWN(g(g(g(g(g(g(g(g(fresh_constant))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) top(up(x0)) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 163 less nodes. ---------------------------------------- (58) Complex Obligation (AND) ---------------------------------------- (59) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(f(g(y4))) -> DOWN(g(y4)) DOWN(g(f(y6))) -> DOWN(f(y6)) DOWN(f(f(g(y10)))) -> DOWN(f(g(y10))) DOWN(f(f(f(g(y16))))) -> DOWN(f(f(g(y16)))) DOWN(f(f(f(f(g(y22)))))) -> DOWN(f(f(f(g(y22))))) DOWN(f(f(f(f(f(g(y28))))))) -> DOWN(f(f(f(f(g(y28)))))) DOWN(f(f(f(f(f(f(g(y34)))))))) -> DOWN(f(f(f(f(f(g(y34))))))) DOWN(f(f(f(f(f(f(f(g(y40))))))))) -> DOWN(f(f(f(f(f(f(g(y40)))))))) DOWN(f(f(f(f(f(f(f(f(g(y46)))))))))) -> DOWN(f(f(f(f(f(f(f(g(y46))))))))) DOWN(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> DOWN(f(f(f(f(f(f(f(f(g(y52)))))))))) DOWN(g(g(f(y12)))) -> DOWN(g(f(y12))) DOWN(g(g(g(f(y18))))) -> DOWN(g(g(f(y18)))) DOWN(g(g(g(g(f(y24)))))) -> DOWN(g(g(g(f(y24))))) DOWN(g(g(g(g(g(f(y30))))))) -> DOWN(g(g(g(g(f(y30)))))) DOWN(g(g(g(g(g(g(f(y36)))))))) -> DOWN(g(g(g(g(g(f(y36))))))) DOWN(g(g(g(g(g(g(g(f(y42))))))))) -> DOWN(g(g(g(g(g(g(f(y42)))))))) DOWN(g(g(g(g(g(g(g(g(f(y48)))))))))) -> DOWN(g(g(g(g(g(g(g(f(y48))))))))) DOWN(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> DOWN(g(g(g(g(g(g(g(g(f(y54)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) top(up(x0)) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (60) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (61) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(f(g(y4))) -> DOWN(g(y4)) DOWN(g(f(y6))) -> DOWN(f(y6)) DOWN(f(f(g(y10)))) -> DOWN(f(g(y10))) DOWN(f(f(f(g(y16))))) -> DOWN(f(f(g(y16)))) DOWN(f(f(f(f(g(y22)))))) -> DOWN(f(f(f(g(y22))))) DOWN(f(f(f(f(f(g(y28))))))) -> DOWN(f(f(f(f(g(y28)))))) DOWN(f(f(f(f(f(f(g(y34)))))))) -> DOWN(f(f(f(f(f(g(y34))))))) DOWN(f(f(f(f(f(f(f(g(y40))))))))) -> DOWN(f(f(f(f(f(f(g(y40)))))))) DOWN(f(f(f(f(f(f(f(f(g(y46)))))))))) -> DOWN(f(f(f(f(f(f(f(g(y46))))))))) DOWN(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> DOWN(f(f(f(f(f(f(f(f(g(y52)))))))))) DOWN(g(g(f(y12)))) -> DOWN(g(f(y12))) DOWN(g(g(g(f(y18))))) -> DOWN(g(g(f(y18)))) DOWN(g(g(g(g(f(y24)))))) -> DOWN(g(g(g(f(y24))))) DOWN(g(g(g(g(g(f(y30))))))) -> DOWN(g(g(g(g(f(y30)))))) DOWN(g(g(g(g(g(g(f(y36)))))))) -> DOWN(g(g(g(g(g(f(y36))))))) DOWN(g(g(g(g(g(g(g(f(y42))))))))) -> DOWN(g(g(g(g(g(g(f(y42)))))))) DOWN(g(g(g(g(g(g(g(g(f(y48)))))))))) -> DOWN(g(g(g(g(g(g(g(f(y48))))))))) DOWN(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> DOWN(g(g(g(g(g(g(g(g(f(y54)))))))))) R is empty. The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) top(up(x0)) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (62) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) top(up(x0)) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) ---------------------------------------- (63) Obligation: Q DP problem: The TRS P consists of the following rules: DOWN(f(g(y4))) -> DOWN(g(y4)) DOWN(g(f(y6))) -> DOWN(f(y6)) DOWN(f(f(g(y10)))) -> DOWN(f(g(y10))) DOWN(f(f(f(g(y16))))) -> DOWN(f(f(g(y16)))) DOWN(f(f(f(f(g(y22)))))) -> DOWN(f(f(f(g(y22))))) DOWN(f(f(f(f(f(g(y28))))))) -> DOWN(f(f(f(f(g(y28)))))) DOWN(f(f(f(f(f(f(g(y34)))))))) -> DOWN(f(f(f(f(f(g(y34))))))) DOWN(f(f(f(f(f(f(f(g(y40))))))))) -> DOWN(f(f(f(f(f(f(g(y40)))))))) DOWN(f(f(f(f(f(f(f(f(g(y46)))))))))) -> DOWN(f(f(f(f(f(f(f(g(y46))))))))) DOWN(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> DOWN(f(f(f(f(f(f(f(f(g(y52)))))))))) DOWN(g(g(f(y12)))) -> DOWN(g(f(y12))) DOWN(g(g(g(f(y18))))) -> DOWN(g(g(f(y18)))) DOWN(g(g(g(g(f(y24)))))) -> DOWN(g(g(g(f(y24))))) DOWN(g(g(g(g(g(f(y30))))))) -> DOWN(g(g(g(g(f(y30)))))) DOWN(g(g(g(g(g(g(f(y36)))))))) -> DOWN(g(g(g(g(g(f(y36))))))) DOWN(g(g(g(g(g(g(g(f(y42))))))))) -> DOWN(g(g(g(g(g(g(f(y42)))))))) DOWN(g(g(g(g(g(g(g(g(f(y48)))))))))) -> DOWN(g(g(g(g(g(g(g(f(y48))))))))) DOWN(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> DOWN(g(g(g(g(g(g(g(g(f(y54)))))))))) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (64) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *DOWN(g(f(y6))) -> DOWN(f(y6)) The graph contains the following edges 1 > 1 *DOWN(f(f(g(y10)))) -> DOWN(f(g(y10))) The graph contains the following edges 1 > 1 *DOWN(g(g(f(y12)))) -> DOWN(g(f(y12))) The graph contains the following edges 1 > 1 *DOWN(f(g(y4))) -> DOWN(g(y4)) The graph contains the following edges 1 > 1 *DOWN(f(f(f(g(y16))))) -> DOWN(f(f(g(y16)))) The graph contains the following edges 1 > 1 *DOWN(f(f(f(f(g(y22)))))) -> DOWN(f(f(f(g(y22))))) The graph contains the following edges 1 > 1 *DOWN(f(f(f(f(f(g(y28))))))) -> DOWN(f(f(f(f(g(y28)))))) The graph contains the following edges 1 > 1 *DOWN(f(f(f(f(f(f(g(y34)))))))) -> DOWN(f(f(f(f(f(g(y34))))))) The graph contains the following edges 1 > 1 *DOWN(f(f(f(f(f(f(f(g(y40))))))))) -> DOWN(f(f(f(f(f(f(g(y40)))))))) The graph contains the following edges 1 > 1 *DOWN(f(f(f(f(f(f(f(f(g(y46)))))))))) -> DOWN(f(f(f(f(f(f(f(g(y46))))))))) The graph contains the following edges 1 > 1 *DOWN(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> DOWN(f(f(f(f(f(f(f(f(g(y52)))))))))) The graph contains the following edges 1 > 1 *DOWN(g(g(g(f(y18))))) -> DOWN(g(g(f(y18)))) The graph contains the following edges 1 > 1 *DOWN(g(g(g(g(f(y24)))))) -> DOWN(g(g(g(f(y24))))) The graph contains the following edges 1 > 1 *DOWN(g(g(g(g(g(f(y30))))))) -> DOWN(g(g(g(g(f(y30)))))) The graph contains the following edges 1 > 1 *DOWN(g(g(g(g(g(g(f(y36)))))))) -> DOWN(g(g(g(g(g(f(y36))))))) The graph contains the following edges 1 > 1 *DOWN(g(g(g(g(g(g(g(f(y42))))))))) -> DOWN(g(g(g(g(g(g(f(y42)))))))) The graph contains the following edges 1 > 1 *DOWN(g(g(g(g(g(g(g(g(f(y48)))))))))) -> DOWN(g(g(g(g(g(g(g(f(y48))))))))) The graph contains the following edges 1 > 1 *DOWN(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> DOWN(g(g(g(g(g(g(g(g(f(y54)))))))))) The graph contains the following edges 1 > 1 ---------------------------------------- (65) YES ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) top(up(x)) -> top(down(x)) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x_1)) -> up(f(x_1)) g_flat(up(x_1)) -> up(g(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) top(up(x0)) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) top(up(x0)) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. top(up(x0)) ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(x)) -> TOP(down(x)) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule TOP(up(x)) -> TOP(down(x)) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(a)) -> TOP(up(f(a))),TOP(up(a)) -> TOP(up(f(a)))) (TOP(up(b)) -> TOP(up(g(b))),TOP(up(b)) -> TOP(up(g(b)))) (TOP(up(f(f(f(f(f(f(f(f(f(f(x0)))))))))))) -> TOP(up(c)),TOP(up(f(f(f(f(f(f(f(f(f(f(x0)))))))))))) -> TOP(up(c))) (TOP(up(g(g(g(g(g(g(g(g(g(g(x0)))))))))))) -> TOP(up(c)),TOP(up(g(g(g(g(g(g(g(g(g(g(x0)))))))))))) -> TOP(up(c))) (TOP(up(f(a))) -> TOP(f_flat(down(a))),TOP(up(f(a))) -> TOP(f_flat(down(a)))) (TOP(up(f(b))) -> TOP(f_flat(down(b))),TOP(up(f(b))) -> TOP(f_flat(down(b)))) (TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))),TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0))))) (TOP(up(f(c))) -> TOP(f_flat(down(c))),TOP(up(f(c))) -> TOP(f_flat(down(c)))) (TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))),TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant)))) (TOP(up(g(a))) -> TOP(g_flat(down(a))),TOP(up(g(a))) -> TOP(g_flat(down(a)))) (TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))),TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0))))) (TOP(up(g(b))) -> TOP(g_flat(down(b))),TOP(up(g(b))) -> TOP(g_flat(down(b)))) (TOP(up(g(c))) -> TOP(g_flat(down(c))),TOP(up(g(c))) -> TOP(g_flat(down(c)))) (TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))),TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant)))) (TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))),TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a))))) (TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))),TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b))))) (TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))),TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0)))))) (TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))),TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c))))) (TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))),TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant))))) (TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))),TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a))))) (TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))),TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0)))))) (TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))),TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b))))) (TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))),TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c))))) (TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))),TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant))))) (TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a)))))) (TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b)))))) (TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))),TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0))))))) (TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))),TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c)))))) (TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))),TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant)))))) (TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))),TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a)))))) (TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))),TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0))))))) (TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))),TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b)))))) (TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))),TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c)))))) (TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))),TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant)))))) (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a))))))) (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b))))))) (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0)))))))) (TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))),TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c))))))) (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant))))))) (TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))),TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a))))))) (TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))),TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0)))))))) (TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))),TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b))))))) (TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))),TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c))))))) (TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))),TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant))))))) (TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))),TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a)))))))) (TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))),TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b)))))))) (TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))),TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0))))))))) (TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))),TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c)))))))) (TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))),TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant)))))))) (TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))),TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a)))))))) (TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))),TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0))))))))) (TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))),TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b)))))))) (TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))),TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c)))))))) (TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))),TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant)))))))) (TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))),TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a))))))))) (TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))),TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b))))))))) (TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))),TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0)))))))))) (TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))),TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c))))))))) (TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))),TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant))))))))) (TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))),TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a))))))))) (TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))),TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0)))))))))) (TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))),TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b))))))))) (TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))),TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c))))))))) (TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))),TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant))))))))) (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a)))))))))) (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b)))))))))) (TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))),TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) (TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))),TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c)))))))))) (TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))),TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a)))))))))) (TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))),TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b)))))))))) (TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))),TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c)))))))))) (TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))),TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a))))))))))) (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b))))))))))) (TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))),TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) (TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))),TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c))))))))))) (TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))),TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a))))))))))) (TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))),TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b))))))))))) (TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))),TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c))))))))))) (TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))),TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a)))))))))))) (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b)))))))))))) (TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0))))))))))))) (TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c)))))))))))) (TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))))) (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a)))))))))))) (TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0))))))))))))) (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b)))))))))))) (TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c)))))))))))) (TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))))) ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(a)) -> TOP(up(f(a))) TOP(up(b)) -> TOP(up(g(b))) TOP(up(f(f(f(f(f(f(f(f(f(f(x0)))))))))))) -> TOP(up(c)) TOP(up(g(g(g(g(g(g(g(g(g(g(x0)))))))))))) -> TOP(up(c)) TOP(up(f(a))) -> TOP(f_flat(down(a))) TOP(up(f(b))) -> TOP(f_flat(down(b))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(f(c))) -> TOP(f_flat(down(c))) TOP(up(f(fresh_constant))) -> TOP(f_flat(down(fresh_constant))) TOP(up(g(a))) -> TOP(g_flat(down(a))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(b))) -> TOP(g_flat(down(b))) TOP(up(g(c))) -> TOP(g_flat(down(c))) TOP(up(g(fresh_constant))) -> TOP(g_flat(down(fresh_constant))) TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes. ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(a))) -> TOP(f_flat(down(a))) TOP(up(f(b))) -> TOP(f_flat(down(b))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(a))) -> TOP(g_flat(down(a))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(b))) -> TOP(g_flat(down(b))) TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (75) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(a))) -> TOP(f_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(a))) -> TOP(f_flat(up(f(a)))),TOP(up(f(a))) -> TOP(f_flat(up(f(a))))) ---------------------------------------- (76) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(b))) -> TOP(f_flat(down(b))) TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(a))) -> TOP(g_flat(down(a))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(b))) -> TOP(g_flat(down(b))) TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (77) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(b))) -> TOP(f_flat(down(b))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(b))) -> TOP(f_flat(up(g(b)))),TOP(up(f(b))) -> TOP(f_flat(up(g(b))))) ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(a))) -> TOP(g_flat(down(a))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(b))) -> TOP(g_flat(down(b))) TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(a))) -> TOP(g_flat(down(a))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(a))) -> TOP(g_flat(up(f(a)))),TOP(up(g(a))) -> TOP(g_flat(up(f(a))))) ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(b))) -> TOP(g_flat(down(b))) TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(b))) -> TOP(g_flat(down(b))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(b))) -> TOP(g_flat(up(g(b)))),TOP(up(g(b))) -> TOP(g_flat(up(g(b))))) ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(a)))) -> TOP(g_flat(down(g(a)))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))),TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a))))) ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(f(x0))))) -> TOP(g_flat(down(g(f(x0))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))),TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0)))))) ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (87) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(down(f(a)))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))),TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a))))) ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(b)))) -> TOP(f_flat(down(f(b)))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))),TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b))))) ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(g(x0))))) -> TOP(f_flat(down(f(g(x0))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))),TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0)))))) ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(c)))) -> TOP(f_flat(down(f(c)))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(c)))) -> TOP(f_flat(f_flat(down(c)))),TOP(up(f(f(c)))) -> TOP(f_flat(f_flat(down(c))))) ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(c)))) -> TOP(f_flat(f_flat(down(c)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (97) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(down(f(fresh_constant)))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant)))),TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant))))) ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(fresh_constant)))) -> TOP(f_flat(f_flat(down(fresh_constant)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (99) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(b)))) -> TOP(g_flat(down(g(b)))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))),TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b))))) ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(c)))) -> TOP(g_flat(down(g(c)))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(c)))) -> TOP(g_flat(g_flat(down(c)))),TOP(up(g(g(c)))) -> TOP(g_flat(g_flat(down(c))))) ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(g(g(c)))) -> TOP(g_flat(g_flat(down(c)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(down(g(fresh_constant)))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(g_flat(down(fresh_constant)))),TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(g_flat(down(fresh_constant))))) ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(g(g(fresh_constant)))) -> TOP(g_flat(g_flat(down(fresh_constant)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(down(f(f(a))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a)))))) ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (113) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(b))))) -> TOP(f_flat(down(f(f(b))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b)))))) ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (115) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(down(f(f(g(x0)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))),TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0))))))) ---------------------------------------- (116) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (117) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(c))))) -> TOP(f_flat(down(f(f(c))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))),TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c)))))) ---------------------------------------- (118) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (119) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(down(f(f(fresh_constant))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))),TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant)))))) ---------------------------------------- (120) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (121) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(down(f(f(f(a)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a))))))) ---------------------------------------- (122) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (123) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(down(f(f(f(b)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b))))))) ---------------------------------------- (124) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (125) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(down(f(f(f(g(x0))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0)))))))) ---------------------------------------- (126) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (127) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(a))))) -> TOP(g_flat(down(g(g(a))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))),TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a)))))) ---------------------------------------- (128) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (129) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(down(g(g(f(x0)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))),TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0))))))) ---------------------------------------- (130) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (131) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(b))))) -> TOP(g_flat(down(g(g(b))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))),TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b)))))) ---------------------------------------- (132) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (133) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(c))))) -> TOP(g_flat(down(g(g(c))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))),TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c)))))) ---------------------------------------- (134) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (135) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(down(g(g(fresh_constant))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))),TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant)))))) ---------------------------------------- (136) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (137) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(down(g(g(g(a)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))),TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a))))))) ---------------------------------------- (138) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (139) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(down(g(g(g(f(x0))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))),TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0)))))))) ---------------------------------------- (140) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (141) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(down(f(f(f(c)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))),TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c))))))) ---------------------------------------- (142) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (143) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(down(f(f(f(fresh_constant)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant))))))) ---------------------------------------- (144) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (145) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(down(f(f(f(f(a))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))),TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a)))))))) ---------------------------------------- (146) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (147) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(down(f(f(f(f(b))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))),TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b)))))))) ---------------------------------------- (148) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (149) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(down(f(f(f(f(g(x0)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))),TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0))))))))) ---------------------------------------- (150) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (151) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(down(g(g(g(b)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))),TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b))))))) ---------------------------------------- (152) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (153) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(down(g(g(g(c)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))),TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c))))))) ---------------------------------------- (154) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (155) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(down(g(g(g(fresh_constant)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))),TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant))))))) ---------------------------------------- (156) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (157) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(down(g(g(g(g(a))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))),TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a)))))))) ---------------------------------------- (158) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (159) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(down(g(g(g(g(f(x0)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))),TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0))))))))) ---------------------------------------- (160) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (161) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(down(f(f(f(f(c))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))),TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c)))))))) ---------------------------------------- (162) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (163) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(down(f(f(f(f(fresh_constant))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))),TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) ---------------------------------------- (164) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (165) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(down(f(f(f(f(f(a)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))),TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a))))))))) ---------------------------------------- (166) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (167) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(down(f(f(f(f(f(b)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))),TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b))))))))) ---------------------------------------- (168) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (169) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(down(f(f(f(f(f(g(x0))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))),TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) ---------------------------------------- (170) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (171) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(down(g(g(g(g(b))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))),TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b)))))))) ---------------------------------------- (172) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (173) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(down(g(g(g(g(c))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))),TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c)))))))) ---------------------------------------- (174) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (175) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(down(g(g(g(g(fresh_constant))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))),TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) ---------------------------------------- (176) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (177) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(down(g(g(g(g(g(a)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))),TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a))))))))) ---------------------------------------- (178) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (179) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(down(g(g(g(g(g(f(x0))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))),TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) ---------------------------------------- (180) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (181) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(down(f(f(f(f(f(c)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))),TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c))))))))) ---------------------------------------- (182) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (183) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(down(f(f(f(f(f(fresh_constant)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))),TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) ---------------------------------------- (184) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (185) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(a))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) ---------------------------------------- (186) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (187) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(b))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) ---------------------------------------- (188) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (189) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(g(x0)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))),TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) ---------------------------------------- (190) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (191) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(down(g(g(g(g(g(b)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))),TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b))))))))) ---------------------------------------- (192) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (193) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(down(g(g(g(g(g(c)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))),TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c))))))))) ---------------------------------------- (194) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (195) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(down(g(g(g(g(g(fresh_constant)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))),TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) ---------------------------------------- (196) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (197) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(a))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) ---------------------------------------- (198) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (199) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(f(x0)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))),TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) ---------------------------------------- (200) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (201) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(c))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))),TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) ---------------------------------------- (202) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (203) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))),TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) ---------------------------------------- (204) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (205) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(a)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) ---------------------------------------- (206) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (207) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(b)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) ---------------------------------------- (208) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (209) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))),TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) ---------------------------------------- (210) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (211) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(b))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) ---------------------------------------- (212) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (213) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(c))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))),TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) ---------------------------------------- (214) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (215) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))),TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) ---------------------------------------- (216) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (217) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(a)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) ---------------------------------------- (218) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (219) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))),TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) ---------------------------------------- (220) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (221) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(c)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))),TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) ---------------------------------------- (222) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (223) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))),TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) ---------------------------------------- (224) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (225) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(a))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a)))))))))))) ---------------------------------------- (226) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (227) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(b))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b)))))))))))) ---------------------------------------- (228) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (229) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(g(x0)))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0))))))))))))) ---------------------------------------- (230) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (231) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(b)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) ---------------------------------------- (232) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (233) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(c)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))),TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) ---------------------------------------- (234) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (235) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))),TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) ---------------------------------------- (236) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (237) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(a))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a)))))))))))) ---------------------------------------- (238) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (239) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(f(x0)))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0))))))))))))) ---------------------------------------- (240) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (241) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(c))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c)))))))))))) ---------------------------------------- (242) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (243) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(down(f(f(f(f(f(f(f(f(fresh_constant))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant)))))))))))) ---------------------------------------- (244) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (245) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(a))) -> TOP(f_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(a))) -> TOP(up(f(f(a)))),TOP(up(f(a))) -> TOP(up(f(f(a))))) ---------------------------------------- (246) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (247) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(b))) -> TOP(f_flat(up(g(b)))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(b))) -> TOP(up(f(g(b)))),TOP(up(f(b))) -> TOP(up(f(g(b))))) ---------------------------------------- (248) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (249) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(down(a)))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))),TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a)))))) ---------------------------------------- (250) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (251) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(down(b)))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))),TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b)))))) ---------------------------------------- (252) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (253) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(b))) -> TOP(g_flat(up(g(b)))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(b))) -> TOP(up(g(g(b)))),TOP(up(g(b))) -> TOP(up(g(g(b))))) ---------------------------------------- (254) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (255) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(b))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b)))))))))))) ---------------------------------------- (256) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (257) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(c))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c)))))))))))) ---------------------------------------- (258) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (259) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(down(g(g(g(g(g(g(g(g(fresh_constant))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant)))))))))))) ---------------------------------------- (260) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (261) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(a))) -> TOP(g_flat(up(f(a)))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(a))) -> TOP(up(g(f(a)))),TOP(up(g(a))) -> TOP(up(g(f(a))))) ---------------------------------------- (262) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (263) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(down(a)))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))),TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a)))))) ---------------------------------------- (264) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (265) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(down(b)))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))),TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b)))))) ---------------------------------------- (266) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (267) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(down(f(a))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a)))))) ---------------------------------------- (268) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (269) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(down(f(b))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b)))))) ---------------------------------------- (270) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (271) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(down(f(g(x0)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))),TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0))))))) ---------------------------------------- (272) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (273) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(down(f(c))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(f_flat(down(c))))),TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(f_flat(down(c)))))) ---------------------------------------- (274) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(c))))) -> TOP(f_flat(f_flat(f_flat(down(c))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (275) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (276) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (277) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(down(f(fresh_constant))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(f_flat(down(fresh_constant))))),TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(f_flat(down(fresh_constant)))))) ---------------------------------------- (278) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(fresh_constant))))) -> TOP(f_flat(f_flat(f_flat(down(fresh_constant))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (279) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (280) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (281) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(down(f(f(a)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a))))))) ---------------------------------------- (282) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (283) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(down(f(f(b)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b))))))) ---------------------------------------- (284) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (285) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(down(f(f(g(x0))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0)))))))) ---------------------------------------- (286) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (287) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(down(g(a))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))),TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a)))))) ---------------------------------------- (288) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (289) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(down(g(f(x0)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))),TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0))))))) ---------------------------------------- (290) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (291) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(down(g(b))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))),TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b)))))) ---------------------------------------- (292) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (293) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(down(g(c))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(g_flat(down(c))))),TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(g_flat(down(c)))))) ---------------------------------------- (294) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(c))))) -> TOP(g_flat(g_flat(g_flat(down(c))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (295) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (296) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (297) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(down(g(fresh_constant))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(g_flat(down(fresh_constant))))),TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(g_flat(down(fresh_constant)))))) ---------------------------------------- (298) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(fresh_constant))))) -> TOP(g_flat(g_flat(g_flat(down(fresh_constant))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (299) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (300) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (301) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(down(g(g(a)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))),TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a))))))) ---------------------------------------- (302) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (303) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(down(g(g(f(x0))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))),TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0)))))))) ---------------------------------------- (304) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (305) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(down(f(f(c)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))),TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c))))))) ---------------------------------------- (306) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (307) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(down(f(f(fresh_constant)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) ---------------------------------------- (308) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (309) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(down(f(f(f(a))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))),TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a)))))))) ---------------------------------------- (310) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (311) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(down(f(f(f(b))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))),TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b)))))))) ---------------------------------------- (312) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (313) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(down(f(f(f(g(x0)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))),TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) ---------------------------------------- (314) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (315) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(down(g(g(b)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))),TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b))))))) ---------------------------------------- (316) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (317) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(down(g(g(c)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))),TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c))))))) ---------------------------------------- (318) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (319) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(down(g(g(fresh_constant)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))),TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) ---------------------------------------- (320) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (321) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(down(g(g(g(a))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))),TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a)))))))) ---------------------------------------- (322) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (323) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(down(g(g(g(f(x0)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))),TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) ---------------------------------------- (324) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (325) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(down(f(f(f(c))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))),TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c)))))))) ---------------------------------------- (326) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (327) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(down(f(f(f(fresh_constant))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))),TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) ---------------------------------------- (328) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (329) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(a)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))),TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) ---------------------------------------- (330) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (331) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(b)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))),TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) ---------------------------------------- (332) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (333) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(g(x0))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))),TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) ---------------------------------------- (334) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (335) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(down(g(g(g(b))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))),TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b)))))))) ---------------------------------------- (336) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (337) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(down(g(g(g(c))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))),TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c)))))))) ---------------------------------------- (338) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (339) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(down(g(g(g(fresh_constant))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))),TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) ---------------------------------------- (340) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (341) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(a)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))),TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) ---------------------------------------- (342) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (343) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(f(x0))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))),TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) ---------------------------------------- (344) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (345) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(c)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))),TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) ---------------------------------------- (346) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (347) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))),TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) ---------------------------------------- (348) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (349) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(a))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) ---------------------------------------- (350) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (351) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(b))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) ---------------------------------------- (352) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (353) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))),TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) ---------------------------------------- (354) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (355) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(b)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))),TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) ---------------------------------------- (356) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (357) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(c)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))),TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) ---------------------------------------- (358) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (359) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))),TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) ---------------------------------------- (360) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (361) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(a))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) ---------------------------------------- (362) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (363) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))),TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) ---------------------------------------- (364) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (365) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(c))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))),TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) ---------------------------------------- (366) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (367) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))),TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) ---------------------------------------- (368) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (369) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) ---------------------------------------- (370) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (371) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) ---------------------------------------- (372) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (373) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))),TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) ---------------------------------------- (374) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (375) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(b))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) ---------------------------------------- (376) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (377) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(c))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))),TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) ---------------------------------------- (378) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (379) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))),TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) ---------------------------------------- (380) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (381) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) ---------------------------------------- (382) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (383) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))),TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) ---------------------------------------- (384) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (385) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))),TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) ---------------------------------------- (386) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (387) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))),TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) ---------------------------------------- (388) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (389) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(a))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a)))))))))))) ---------------------------------------- (390) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (391) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(b))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b)))))))))))) ---------------------------------------- (392) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (393) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(g(x0)))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0))))))))))))) ---------------------------------------- (394) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (395) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) ---------------------------------------- (396) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (397) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))),TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) ---------------------------------------- (398) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (399) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))),TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) ---------------------------------------- (400) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (401) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(a))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a)))))))))))) ---------------------------------------- (402) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (403) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(f(x0)))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0))))))))))))) ---------------------------------------- (404) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (405) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(c))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c)))))))))))) ---------------------------------------- (406) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (407) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant)))))))))))) ---------------------------------------- (408) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (409) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(f_flat(up(f(a))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))),TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a)))))) ---------------------------------------- (410) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (411) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(b)))) -> TOP(f_flat(f_flat(up(g(b))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))),TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b)))))) ---------------------------------------- (412) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (413) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(down(a))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a))))))) ---------------------------------------- (414) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (415) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(down(b))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b))))))) ---------------------------------------- (416) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (417) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(b)))) -> TOP(g_flat(g_flat(up(g(b))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))),TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b)))))) ---------------------------------------- (418) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (419) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(b))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b)))))))))))) ---------------------------------------- (420) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (421) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(c))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c)))))))))))) ---------------------------------------- (422) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (423) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant)))))))))))) ---------------------------------------- (424) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (425) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(a)))) -> TOP(g_flat(g_flat(up(f(a))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))),TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a)))))) ---------------------------------------- (426) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (427) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(down(a))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))),TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a))))))) ---------------------------------------- (428) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (429) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(down(f(a)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a))))))) ---------------------------------------- (430) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (431) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(down(f(b)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b))))))) ---------------------------------------- (432) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (433) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(down(f(g(x0))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))),TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) ---------------------------------------- (434) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (435) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(down(b))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(up(g(b)))))),TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(up(g(b))))))) ---------------------------------------- (436) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(up(g(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (437) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(down(g(a)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))),TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a))))))) ---------------------------------------- (438) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(up(g(b)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (439) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(down(g(f(x0))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))),TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) ---------------------------------------- (440) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(up(g(b)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (441) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(down(f(c)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(c)))))),TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(c))))))) ---------------------------------------- (442) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(up(g(b)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(c)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(c)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (443) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (444) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(up(g(b)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (445) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(a)))) -> TOP(f_flat(up(f(f(a))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))),TOP(up(f(f(a)))) -> TOP(up(f(f(f(a)))))) ---------------------------------------- (446) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(up(g(b)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (447) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(b)))) -> TOP(g_flat(up(g(g(b))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))),TOP(up(g(g(b)))) -> TOP(up(g(g(g(b)))))) ---------------------------------------- (448) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(up(g(b)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (449) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(down(f(fresh_constant)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))),TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(fresh_constant))))))) ---------------------------------------- (450) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(up(g(b)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(f(f(f(f(fresh_constant)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (451) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (452) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(up(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (453) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(f_flat(up(f(a)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a))))))) ---------------------------------------- (454) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(up(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (455) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(g_flat(up(g(b)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(up(g(g(b)))))),TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(up(g(g(b))))))) ---------------------------------------- (456) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(up(g(g(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (457) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(a))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))),TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) ---------------------------------------- (458) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(up(g(g(b)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (459) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(b))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))),TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) ---------------------------------------- (460) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(up(g(g(b)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (461) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))),TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) ---------------------------------------- (462) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(up(g(g(b)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (463) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(down(g(b)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))),TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b))))))) ---------------------------------------- (464) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(up(g(g(b)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (465) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(down(g(c)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(c)))))),TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(c))))))) ---------------------------------------- (466) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(up(g(g(b)))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(g(g(g(g(c)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(c)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (467) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (468) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(up(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (469) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(f_flat(up(f(f(a)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a))))))) ---------------------------------------- (470) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(up(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (471) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(b))))) -> TOP(g_flat(g_flat(up(g(g(b)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(b))))) -> TOP(g_flat(up(g(g(g(b)))))),TOP(up(g(g(g(b))))) -> TOP(g_flat(up(g(g(g(b))))))) ---------------------------------------- (472) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(up(g(g(g(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (473) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(down(g(fresh_constant)))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(fresh_constant)))))),TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(fresh_constant))))))) ---------------------------------------- (474) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(up(g(g(g(b)))))) TOP(up(g(g(g(g(fresh_constant)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(fresh_constant)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (475) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (476) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(up(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (477) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(a))))) -> TOP(f_flat(up(f(f(f(a)))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))),TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a))))))) ---------------------------------------- (478) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(g_flat(up(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (479) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(b))))) -> TOP(g_flat(up(g(g(g(b)))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))),TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b))))))) ---------------------------------------- (480) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (481) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(a))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))),TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) ---------------------------------------- (482) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (483) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))),TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) ---------------------------------------- (484) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (485) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(c))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))),TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) ---------------------------------------- (486) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (487) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))),TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) ---------------------------------------- (488) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (489) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(a)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))),TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) ---------------------------------------- (490) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (491) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(b)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))),TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) ---------------------------------------- (492) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (493) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))),TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) ---------------------------------------- (494) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (495) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(b))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))),TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) ---------------------------------------- (496) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (497) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(c))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))),TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) ---------------------------------------- (498) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (499) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))),TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) ---------------------------------------- (500) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (501) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(a)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))),TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) ---------------------------------------- (502) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (503) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))),TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) ---------------------------------------- (504) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (505) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(c)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))),TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) ---------------------------------------- (506) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (507) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))),TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) ---------------------------------------- (508) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (509) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) ---------------------------------------- (510) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (511) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) ---------------------------------------- (512) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (513) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))),TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) ---------------------------------------- (514) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (515) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(b)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))),TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) ---------------------------------------- (516) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (517) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(c)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))),TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) ---------------------------------------- (518) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (519) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))),TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) ---------------------------------------- (520) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (521) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) ---------------------------------------- (522) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (523) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))),TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) ---------------------------------------- (524) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (525) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))),TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) ---------------------------------------- (526) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (527) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))),TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) ---------------------------------------- (528) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (529) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) ---------------------------------------- (530) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (531) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) ---------------------------------------- (532) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (533) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))),TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) ---------------------------------------- (534) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (535) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) ---------------------------------------- (536) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (537) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))),TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) ---------------------------------------- (538) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (539) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))),TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) ---------------------------------------- (540) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (541) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) ---------------------------------------- (542) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (543) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))),TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) ---------------------------------------- (544) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (545) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))),TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) ---------------------------------------- (546) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (547) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))),TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) ---------------------------------------- (548) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (549) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(a))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a)))))))))))) ---------------------------------------- (550) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (551) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(b))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b)))))))))))) ---------------------------------------- (552) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (553) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(g(x0)))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0))))))))))))) ---------------------------------------- (554) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (555) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) ---------------------------------------- (556) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (557) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))),TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) ---------------------------------------- (558) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (559) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))),TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) ---------------------------------------- (560) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (561) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(a))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a)))))))))))) ---------------------------------------- (562) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (563) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(f(x0)))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0))))))))))))) ---------------------------------------- (564) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (565) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(c))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c)))))))))))) ---------------------------------------- (566) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (567) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(down(f(f(f(f(f(f(fresh_constant))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant)))))))))))) ---------------------------------------- (568) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (569) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(b)))) -> TOP(f_flat(up(f(g(b))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))),TOP(up(f(f(b)))) -> TOP(up(f(f(g(b)))))) ---------------------------------------- (570) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (571) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(f_flat(up(g(b)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b))))))) ---------------------------------------- (572) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (573) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(a)))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) ---------------------------------------- (574) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (575) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(b)))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) ---------------------------------------- (576) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (577) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(b))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b)))))))))))) ---------------------------------------- (578) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (579) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(c))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c)))))))))))) ---------------------------------------- (580) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (581) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(down(g(g(g(g(g(g(fresh_constant))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant)))))))))))) ---------------------------------------- (582) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (583) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(a)))) -> TOP(g_flat(up(g(f(a))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))),TOP(up(g(g(a)))) -> TOP(up(g(g(f(a)))))) ---------------------------------------- (584) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (585) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(g_flat(up(f(a)))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))),TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a))))))) ---------------------------------------- (586) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (587) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(a)))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))),TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) ---------------------------------------- (588) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (589) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(a))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))),TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) ---------------------------------------- (590) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (591) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(b))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))),TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) ---------------------------------------- (592) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (593) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))),TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) ---------------------------------------- (594) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (595) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(b)))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(b))))))),TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))) ---------------------------------------- (596) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (597) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(a))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))),TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) ---------------------------------------- (598) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(b))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (599) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))),TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) ---------------------------------------- (600) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(b))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (601) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(c))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(c))))))),TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(c)))))))) ---------------------------------------- (602) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(b))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(c))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(c))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (603) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (604) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (605) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(a))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a)))))))) ---------------------------------------- (606) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (607) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(b))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(b))))))),TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(b)))))))) ---------------------------------------- (608) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (609) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant))))))),TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))))) ---------------------------------------- (610) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(b))))))) TOP(up(f(f(f(f(f(fresh_constant))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (611) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (612) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (613) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(a))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a)))))))) ---------------------------------------- (614) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (615) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(b))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(up(g(g(g(b))))))),TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(up(g(g(g(b)))))))) ---------------------------------------- (616) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(up(g(g(g(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (617) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))),TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) ---------------------------------------- (618) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(up(g(g(g(b))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (619) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))),TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) ---------------------------------------- (620) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(up(g(g(g(b))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (621) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))),TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) ---------------------------------------- (622) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(up(g(g(g(b))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (623) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(b))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))),TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) ---------------------------------------- (624) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(up(g(g(g(b))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (625) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(c))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(c))))))),TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(c)))))))) ---------------------------------------- (626) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(up(g(g(g(b))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(g(g(g(g(g(c))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(c))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (627) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (628) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(up(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (629) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(f_flat(up(f(f(f(a))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a)))))))) ---------------------------------------- (630) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(up(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (631) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(g_flat(up(g(g(g(b))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(up(g(g(g(g(b))))))),TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(up(g(g(g(g(b)))))))) ---------------------------------------- (632) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(up(g(g(g(g(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (633) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant))))))),TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant)))))))) ---------------------------------------- (634) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(up(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(fresh_constant))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (635) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (636) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(up(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (637) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(a)))))) -> TOP(f_flat(up(f(f(f(f(a))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))),TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a)))))))) ---------------------------------------- (638) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(up(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (639) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(b)))))) -> TOP(g_flat(up(g(g(g(g(b))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))),TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b)))))))) ---------------------------------------- (640) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (641) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))),TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) ---------------------------------------- (642) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (643) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))),TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) ---------------------------------------- (644) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (645) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))),TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) ---------------------------------------- (646) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (647) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))),TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) ---------------------------------------- (648) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (649) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) ---------------------------------------- (650) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (651) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) ---------------------------------------- (652) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (653) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))),TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) ---------------------------------------- (654) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (655) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))),TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) ---------------------------------------- (656) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (657) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))),TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) ---------------------------------------- (658) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (659) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))),TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) ---------------------------------------- (660) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (661) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) ---------------------------------------- (662) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (663) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))),TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) ---------------------------------------- (664) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (665) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))),TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) ---------------------------------------- (666) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (667) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))),TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) ---------------------------------------- (668) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (669) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) ---------------------------------------- (670) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (671) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) ---------------------------------------- (672) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (673) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))),TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) ---------------------------------------- (674) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (675) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) ---------------------------------------- (676) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (677) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))),TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) ---------------------------------------- (678) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (679) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))),TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) ---------------------------------------- (680) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (681) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) ---------------------------------------- (682) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (683) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))),TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) ---------------------------------------- (684) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (685) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))),TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) ---------------------------------------- (686) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (687) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))),TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) ---------------------------------------- (688) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (689) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(a))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a)))))))))))) ---------------------------------------- (690) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (691) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(b))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b)))))))))))) ---------------------------------------- (692) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (693) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(g(x0)))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0))))))))))))) ---------------------------------------- (694) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (695) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) ---------------------------------------- (696) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (697) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))),TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) ---------------------------------------- (698) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (699) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))),TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) ---------------------------------------- (700) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (701) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(a))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a)))))))))))) ---------------------------------------- (702) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (703) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(f(x0)))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0))))))))))))) ---------------------------------------- (704) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (705) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(c))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c)))))))))))) ---------------------------------------- (706) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (707) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(f(fresh_constant))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant)))))))))))) ---------------------------------------- (708) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (709) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(b))))) -> TOP(f_flat(f_flat(up(f(g(b)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))),TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b))))))) ---------------------------------------- (710) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (711) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(g(b))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b)))))))) ---------------------------------------- (712) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (713) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))),TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) ---------------------------------------- (714) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (715) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))),TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) ---------------------------------------- (716) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (717) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(b))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b)))))))))))) ---------------------------------------- (718) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (719) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(c))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c)))))))))))) ---------------------------------------- (720) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (721) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(g(fresh_constant))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant)))))))))))) ---------------------------------------- (722) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (723) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(a))))) -> TOP(g_flat(g_flat(up(g(f(a)))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))),TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a))))))) ---------------------------------------- (724) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (725) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(f(a))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))),TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a)))))))) ---------------------------------------- (726) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (727) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))),TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) ---------------------------------------- (728) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (729) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))),TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) ---------------------------------------- (730) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (731) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))),TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) ---------------------------------------- (732) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (733) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))),TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) ---------------------------------------- (734) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (735) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))),TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))) ---------------------------------------- (736) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (737) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))),TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) ---------------------------------------- (738) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (739) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))),TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) ---------------------------------------- (740) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (741) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(c)))))))),TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(c))))))))) ---------------------------------------- (742) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(c)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(c)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (743) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (744) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (745) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))),TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))) ---------------------------------------- (746) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (747) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))),TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))) ---------------------------------------- (748) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (749) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))))),TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant))))))))) ---------------------------------------- (750) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))) TOP(up(f(f(f(f(f(f(fresh_constant)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (751) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (752) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (753) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(a)))))))),TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(a))))))))) ---------------------------------------- (754) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (755) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(b)))))))),TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(b))))))))) ---------------------------------------- (756) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (757) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) ---------------------------------------- (758) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(b)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (759) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) ---------------------------------------- (760) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(b)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (761) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))),TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) ---------------------------------------- (762) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(b)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (763) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))),TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) ---------------------------------------- (764) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(b)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (765) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(c)))))))),TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(c))))))))) ---------------------------------------- (766) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(b)))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(g(g(g(g(g(g(c)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(c)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (767) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (768) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (769) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(a)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(a)))))))),TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(a))))))))) ---------------------------------------- (770) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (771) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(b)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(b)))))))),TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(b))))))))) ---------------------------------------- (772) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (773) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant)))))))),TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant))))))))) ---------------------------------------- (774) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(fresh_constant)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (775) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (776) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (777) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(a)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))),TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a))))))))) ---------------------------------------- (778) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (779) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(b)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))),TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b))))))))) ---------------------------------------- (780) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (781) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) ---------------------------------------- (782) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (783) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))),TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) ---------------------------------------- (784) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (785) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))),TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) ---------------------------------------- (786) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (787) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))),TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) ---------------------------------------- (788) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (789) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) ---------------------------------------- (790) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (791) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) ---------------------------------------- (792) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (793) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))),TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) ---------------------------------------- (794) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (795) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) ---------------------------------------- (796) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (797) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))),TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) ---------------------------------------- (798) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (799) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))),TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) ---------------------------------------- (800) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (801) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) ---------------------------------------- (802) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (803) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))),TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) ---------------------------------------- (804) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (805) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))),TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) ---------------------------------------- (806) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (807) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))),TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) ---------------------------------------- (808) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (809) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(a))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a)))))))))))) ---------------------------------------- (810) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (811) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(b))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b)))))))))))) ---------------------------------------- (812) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (813) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(g(x0)))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0))))))))))))) ---------------------------------------- (814) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (815) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) ---------------------------------------- (816) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (817) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))),TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) ---------------------------------------- (818) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (819) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))),TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) ---------------------------------------- (820) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (821) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(a))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a)))))))))))) ---------------------------------------- (822) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (823) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(f(x0)))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0))))))))))))) ---------------------------------------- (824) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (825) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(c))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c)))))))))))) ---------------------------------------- (826) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (827) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(f(fresh_constant))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant)))))))))))) ---------------------------------------- (828) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (829) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(b))))) -> TOP(f_flat(up(f(f(g(b)))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))),TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b))))))) ---------------------------------------- (830) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (831) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(f_flat(up(f(g(b))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b)))))))) ---------------------------------------- (832) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (833) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))),TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) ---------------------------------------- (834) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (835) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))),TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) ---------------------------------------- (836) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (837) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))),TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) ---------------------------------------- (838) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (839) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(b))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b)))))))))))) ---------------------------------------- (840) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (841) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(c))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c)))))))))))) ---------------------------------------- (842) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (843) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(g(fresh_constant))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant)))))))))))) ---------------------------------------- (844) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (845) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(a))))) -> TOP(g_flat(up(g(g(f(a)))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))),TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a))))))) ---------------------------------------- (846) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (847) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(g_flat(up(g(f(a))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))),TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a)))))))) ---------------------------------------- (848) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (849) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))),TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) ---------------------------------------- (850) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (851) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))),TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) ---------------------------------------- (852) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (853) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) ---------------------------------------- (854) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (855) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) ---------------------------------------- (856) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (857) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))),TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) ---------------------------------------- (858) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (859) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))),TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) ---------------------------------------- (860) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (861) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(a))))))) -> TOP(f_flat(up(f(f(f(f(f(a)))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))),TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a))))))))) ---------------------------------------- (862) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (863) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(b))))))) -> TOP(g_flat(up(g(g(g(g(g(b)))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))),TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b))))))))) ---------------------------------------- (864) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (865) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) ---------------------------------------- (866) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (867) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))),TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) ---------------------------------------- (868) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (869) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(c))))))))),TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(c)))))))))) ---------------------------------------- (870) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(c))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(c))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (871) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (872) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (873) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))),TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))) ---------------------------------------- (874) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (875) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))),TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))) ---------------------------------------- (876) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (877) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant))))))))),TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))))))) ---------------------------------------- (878) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))) TOP(up(f(f(f(f(f(f(f(fresh_constant))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (879) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (880) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (881) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))),TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))) ---------------------------------------- (882) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (883) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))),TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))) ---------------------------------------- (884) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (885) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) ---------------------------------------- (886) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (887) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) ---------------------------------------- (888) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (889) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))),TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) ---------------------------------------- (890) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (891) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) ---------------------------------------- (892) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (893) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(c))))))))),TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(c)))))))))) ---------------------------------------- (894) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(g(g(g(g(g(g(g(c))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(c))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (895) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (896) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (897) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))),TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) ---------------------------------------- (898) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (899) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))),TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) ---------------------------------------- (900) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (901) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant))))))))),TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant)))))))))) ---------------------------------------- (902) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(fresh_constant))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (903) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (904) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (905) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))),TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a)))))))))) ---------------------------------------- (906) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (907) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))),TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b)))))))))) ---------------------------------------- (908) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (909) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) ---------------------------------------- (910) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (911) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))),TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) ---------------------------------------- (912) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (913) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))),TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) ---------------------------------------- (914) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (915) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))),TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) ---------------------------------------- (916) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (917) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(a))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a)))))))))))) ---------------------------------------- (918) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (919) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(b))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b)))))))))))) ---------------------------------------- (920) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (921) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(g(x0)))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0))))))))))))) ---------------------------------------- (922) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (923) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) ---------------------------------------- (924) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (925) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))),TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) ---------------------------------------- (926) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (927) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))),TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) ---------------------------------------- (928) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (929) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(a))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a)))))))))))) ---------------------------------------- (930) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (931) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(f(x0)))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0))))))))))))) ---------------------------------------- (932) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (933) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(c))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c)))))))))))) ---------------------------------------- (934) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (935) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(f(fresh_constant))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant)))))))))))) ---------------------------------------- (936) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (937) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(f_flat(up(f(f(g(b))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))),TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b)))))))) ---------------------------------------- (938) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (939) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))),TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) ---------------------------------------- (940) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (941) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))),TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) ---------------------------------------- (942) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (943) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) ---------------------------------------- (944) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (945) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) ---------------------------------------- (946) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (947) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(b))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b)))))))))))) ---------------------------------------- (948) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (949) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(c))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c)))))))))))) ---------------------------------------- (950) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (951) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(g(fresh_constant))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant)))))))))))) ---------------------------------------- (952) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (953) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(g_flat(up(g(g(f(a))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))),TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a)))))))) ---------------------------------------- (954) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (955) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))),TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) ---------------------------------------- (956) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (957) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))),TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) ---------------------------------------- (958) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (959) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) ---------------------------------------- (960) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (961) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) ---------------------------------------- (962) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (963) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) ---------------------------------------- (964) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (965) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))),TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) ---------------------------------------- (966) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (967) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) ---------------------------------------- (968) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (969) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(a))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(up(f(f(f(f(f(f(a))))))))),TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(up(f(f(f(f(f(f(a)))))))))) ---------------------------------------- (970) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(up(f(f(f(f(f(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (971) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(b))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(up(g(g(g(g(g(g(b))))))))),TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(up(g(g(g(g(g(g(b)))))))))) ---------------------------------------- (972) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(up(f(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(up(g(g(g(g(g(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (973) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) ---------------------------------------- (974) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(up(f(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(up(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (975) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))),TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) ---------------------------------------- (976) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(up(f(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(up(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (977) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(c)))))))))),TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(c))))))))))) ---------------------------------------- (978) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(up(f(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(up(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(c)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (979) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (980) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(up(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(up(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (981) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(f_flat(up(f(f(f(f(f(f(a))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))),TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a)))))))))) ---------------------------------------- (982) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(up(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (983) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(g_flat(up(g(g(g(g(g(g(b))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))),TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b)))))))))) ---------------------------------------- (984) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (985) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))))))),TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant))))))))))) ---------------------------------------- (986) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (987) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (988) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (989) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))) ---------------------------------------- (990) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (991) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) ---------------------------------------- (992) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (993) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(a))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a)))))))))))) ---------------------------------------- (994) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (995) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(b))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b)))))))))))) ---------------------------------------- (996) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (997) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(g(x0)))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0))))))))))))) ---------------------------------------- (998) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (999) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) ---------------------------------------- (1000) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1001) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(c)))))))))),TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(c))))))))))) ---------------------------------------- (1002) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(c)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(c)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1003) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1004) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1005) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) ---------------------------------------- (1006) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1007) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) ---------------------------------------- (1008) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1009) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant)))))))))),TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant))))))))))) ---------------------------------------- (1010) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1011) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1012) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1013) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) ---------------------------------------- (1014) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1015) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) ---------------------------------------- (1016) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1017) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(a))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a)))))))))))) ---------------------------------------- (1018) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1019) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(f(x0)))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0))))))))))))) ---------------------------------------- (1020) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1021) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(c))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c)))))))))))) ---------------------------------------- (1022) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1023) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(f(fresh_constant))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant)))))))))))) ---------------------------------------- (1024) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1025) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(b)))))) -> TOP(f_flat(up(f(f(f(g(b))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))),TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b)))))))) ---------------------------------------- (1026) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1027) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(g(b)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))),TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b))))))))) ---------------------------------------- (1028) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1029) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))),TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) ---------------------------------------- (1030) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1031) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) ---------------------------------------- (1032) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1033) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) ---------------------------------------- (1034) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1035) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) ---------------------------------------- (1036) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1037) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(b))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b)))))))))))) ---------------------------------------- (1038) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1039) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(c))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c)))))))))))) ---------------------------------------- (1040) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1041) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(g(fresh_constant))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant)))))))))))) ---------------------------------------- (1042) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1043) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(a)))))) -> TOP(g_flat(up(g(g(g(f(a))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))),TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a)))))))) ---------------------------------------- (1044) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1045) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(f(a)))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))),TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a))))))))) ---------------------------------------- (1046) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1047) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))),TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) ---------------------------------------- (1048) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1049) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) ---------------------------------------- (1050) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1051) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) ---------------------------------------- (1052) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1053) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(a))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a)))))))))))) ---------------------------------------- (1054) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1055) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(b))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b)))))))))))) ---------------------------------------- (1056) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1057) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(g(x0)))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))))) ---------------------------------------- (1058) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1059) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) ---------------------------------------- (1060) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1061) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) ---------------------------------------- (1062) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1063) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))) ---------------------------------------- (1064) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1065) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(a))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a)))))))))))) ---------------------------------------- (1066) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1067) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(f(x0)))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))))) ---------------------------------------- (1068) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1069) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(c))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(c))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(c)))))))))))) ---------------------------------------- (1070) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(c))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1071) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1072) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1073) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))) ---------------------------------------- (1074) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1075) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))) ---------------------------------------- (1076) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1077) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(f(fresh_constant))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant)))))))))))) ---------------------------------------- (1078) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(fresh_constant))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1079) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1080) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1081) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a))))))))))) ---------------------------------------- (1082) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1083) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b))))))))))) ---------------------------------------- (1084) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1085) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(f_flat(up(f(f(f(g(b)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))),TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b))))))))) ---------------------------------------- (1086) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1087) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))),TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) ---------------------------------------- (1088) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1089) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) ---------------------------------------- (1090) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1091) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) ---------------------------------------- (1092) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1093) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) ---------------------------------------- (1094) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1095) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(a))))))))))) at position [0,0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a))))))))))))) ---------------------------------------- (1096) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1097) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(b))))))))))) at position [0,0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b))))))))))))) ---------------------------------------- (1098) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1099) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(b))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b)))))))))))) ---------------------------------------- (1100) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1101) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(c))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(c))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(c)))))))))))) ---------------------------------------- (1102) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(c))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(c))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1103) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1104) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1105) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(a)))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))),TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a))))))))))) ---------------------------------------- (1106) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1107) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(b)))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))),TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b))))))))))) ---------------------------------------- (1108) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1109) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(g(fresh_constant))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant)))))))))))) ---------------------------------------- (1110) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(fresh_constant))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(fresh_constant))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1111) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1112) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1113) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) ---------------------------------------- (1114) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1115) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) ---------------------------------------- (1116) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1117) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(g_flat(up(g(g(g(f(a)))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))),TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a))))))))) ---------------------------------------- (1118) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1119) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))),TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) ---------------------------------------- (1120) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1121) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) ---------------------------------------- (1122) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1123) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) ---------------------------------------- (1124) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1125) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(a))))))))))) at position [0,0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a))))))))))))) ---------------------------------------- (1126) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1127) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(b))))))) -> TOP(f_flat(up(f(f(f(f(g(b)))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))),TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b))))))))) ---------------------------------------- (1128) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1129) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))),TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) ---------------------------------------- (1130) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1131) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) ---------------------------------------- (1132) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1133) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) ---------------------------------------- (1134) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1135) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(a)))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a))))))))))))) ---------------------------------------- (1136) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1137) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(g(b)))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b))))))))))))) ---------------------------------------- (1138) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1139) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(b))))))))))) at position [0,0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b))))))))))))) ---------------------------------------- (1140) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1141) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))) ---------------------------------------- (1142) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1143) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))) ---------------------------------------- (1144) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1145) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(a))))))) -> TOP(g_flat(up(g(g(g(g(f(a)))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))),TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a))))))))) ---------------------------------------- (1146) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1147) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))),TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) ---------------------------------------- (1148) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1149) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) ---------------------------------------- (1150) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1151) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) ---------------------------------------- (1152) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1153) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(f(a)))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a))))))))))))) ---------------------------------------- (1154) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1155) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(g(b))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))),TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b)))))))))) ---------------------------------------- (1156) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1157) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))) ---------------------------------------- (1158) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1159) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))) ---------------------------------------- (1160) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1161) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(a)))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a))))))))))))) ---------------------------------------- (1162) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1163) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(g(b)))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b))))))))))))) ---------------------------------------- (1164) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1165) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(b)))))))))))) at position [0,0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b))))))))))))) ---------------------------------------- (1166) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1167) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))))) ---------------------------------------- (1168) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1169) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))))) ---------------------------------------- (1170) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1171) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(f(a))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))),TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a)))))))))) ---------------------------------------- (1172) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1173) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))) ---------------------------------------- (1174) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1175) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))))) ---------------------------------------- (1176) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1177) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(f(a)))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a))))))))))))) ---------------------------------------- (1178) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1179) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(f_flat(up(f(f(f(f(f(g(b))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))),TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b)))))))))) ---------------------------------------- (1180) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1181) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))) ---------------------------------------- (1182) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1183) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))))) ---------------------------------------- (1184) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1185) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(a)))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a))))))))))))) ---------------------------------------- (1186) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1187) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(g(b)))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b))))))))))))) ---------------------------------------- (1188) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1189) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(b)))))))))))) at position [0,0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b))))))))))))) ---------------------------------------- (1190) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1191) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))))) ---------------------------------------- (1192) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1193) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))))) ---------------------------------------- (1194) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1195) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(g_flat(up(g(g(g(g(g(f(a))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))),TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a)))))))))) ---------------------------------------- (1196) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1197) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))) ---------------------------------------- (1198) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1199) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))))) ---------------------------------------- (1200) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1201) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(f(a)))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a))))))))))))) ---------------------------------------- (1202) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1203) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(g(b)))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(g(b))))))))))) ---------------------------------------- (1204) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1205) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))))) ---------------------------------------- (1206) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1207) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(a)))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a))))))))))))) ---------------------------------------- (1208) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1209) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(g(b)))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b))))))))))))) ---------------------------------------- (1210) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1211) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(b)))))))))))) at position [0,0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b))))))))))))) ---------------------------------------- (1212) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1213) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(a)))))))))))) ---------------------------------------- (1214) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1215) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))))) ---------------------------------------- (1216) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1217) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(f(a)))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(f(a))))))))))) ---------------------------------------- (1218) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1219) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))))) ---------------------------------------- (1220) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1221) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(f(a)))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a))))))))))))) ---------------------------------------- (1222) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1223) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(g(b)))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))),TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b))))))))))) ---------------------------------------- (1224) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1225) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(g(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(g(b)))))))))))) ---------------------------------------- (1226) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1227) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(a)))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a))))))))))))) ---------------------------------------- (1228) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1229) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(g(b)))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b))))))))))))) ---------------------------------------- (1230) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1231) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(b)))))))))))) at position [0,0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b))))))))))))) ---------------------------------------- (1232) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1233) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(a))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(a)))))))))))) ---------------------------------------- (1234) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1235) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(b)))))))))))) ---------------------------------------- (1236) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1237) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(f(a)))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))),TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a))))))))))) ---------------------------------------- (1238) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1239) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(f(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(f(a)))))))))))) ---------------------------------------- (1240) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1241) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(f(a)))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a))))))))))))) ---------------------------------------- (1242) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1243) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(g(b))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(g(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(g(b)))))))))))) ---------------------------------------- (1244) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1245) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(a)))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(f(a)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(f(a))))))))))))) ---------------------------------------- (1246) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1247) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(up(f(f(f(f(f(g(b)))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(g(b)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(g(b))))))))))))) ---------------------------------------- (1248) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1249) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(b)))))))))))) at position [0,0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b))))))))))))) ---------------------------------------- (1250) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1251) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(a))))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))),TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a)))))))))))) ---------------------------------------- (1252) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1253) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(b))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(b)))))))))))) ---------------------------------------- (1254) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1255) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(f(a))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(f(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(f(a)))))))))))) ---------------------------------------- (1256) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1257) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(f(a)))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(f(a)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(f(a))))))))))))) ---------------------------------------- (1258) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1259) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(g(b))))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))),TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b)))))))))))) ---------------------------------------- (1260) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1261) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(f(a)))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(f(a)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(f(a))))))))))))) ---------------------------------------- (1262) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1263) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(f_flat(up(f(f(f(f(f(f(g(b)))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(g(b)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(g(b))))))))))))) ---------------------------------------- (1264) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1265) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(b)))))))))))) at position [0,0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(g(b)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(g(b))))))))))))) ---------------------------------------- (1266) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1267) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(b))))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))),TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b)))))))))))) ---------------------------------------- (1268) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1269) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(f(a))))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))),TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a)))))))))))) ---------------------------------------- (1270) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1271) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(f(a)))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(f(a)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(f(a))))))))))))) ---------------------------------------- (1272) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1273) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(f(a)))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(f(a)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(f(a))))))))))))) ---------------------------------------- (1274) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1275) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(f_flat(up(f(f(f(f(f(f(f(g(b)))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(g(b)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(g(b))))))))))))) ---------------------------------------- (1276) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1277) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(g_flat(up(g(g(g(g(g(g(g(b)))))))))))) at position [0,0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(g(b)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(g(b))))))))))))) ---------------------------------------- (1278) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1279) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(f(a)))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(f(a)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(f(a))))))))))))) ---------------------------------------- (1280) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1281) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(f(a)))))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(f(a)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(f(a))))))))))))) ---------------------------------------- (1282) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(a))) -> TOP(up(f(f(a)))) TOP(up(f(f(a)))) -> TOP(up(f(f(f(a))))) TOP(up(f(f(f(a))))) -> TOP(up(f(f(f(f(a)))))) TOP(up(f(f(f(f(a)))))) -> TOP(up(f(f(f(f(f(a))))))) TOP(up(f(f(f(f(f(a))))))) -> TOP(up(f(f(f(f(f(f(a)))))))) TOP(up(f(f(f(f(f(f(a)))))))) -> TOP(up(f(f(f(f(f(f(f(a))))))))) TOP(up(f(f(f(f(f(f(f(a))))))))) -> TOP(up(f(f(f(f(f(f(f(f(a)))))))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(a)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(f(a)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(a))))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1283) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes. ---------------------------------------- (1284) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(g(b)))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1285) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(g_flat(up(g(g(g(g(g(g(g(g(b)))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(g(b)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(g(b))))))))))))) ---------------------------------------- (1286) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(f(a)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1287) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(f_flat(up(f(f(f(f(f(f(f(f(g(b)))))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(g(b)))))))))))),TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(g(b))))))))))))) ---------------------------------------- (1288) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(f(a)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1289) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(f(a)))))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(f(a)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(f(a))))))))))))) ---------------------------------------- (1290) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(g(b)))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1291) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(g_flat(up(g(g(g(g(g(g(g(g(g(b)))))))))))) at position [0] we obtained the following new rules [LPAR04]: (TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(g(b)))))))))))),TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(g(b))))))))))))) ---------------------------------------- (1292) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(b))) -> TOP(up(g(g(b)))) TOP(up(g(g(b)))) -> TOP(up(g(g(g(b))))) TOP(up(g(g(g(b))))) -> TOP(up(g(g(g(g(b)))))) TOP(up(g(g(g(g(b)))))) -> TOP(up(g(g(g(g(g(b))))))) TOP(up(g(g(g(g(g(b))))))) -> TOP(up(g(g(g(g(g(g(b)))))))) TOP(up(g(g(g(g(g(g(b)))))))) -> TOP(up(g(g(g(g(g(g(g(b))))))))) TOP(up(g(g(g(g(g(g(g(b))))))))) -> TOP(up(g(g(g(g(g(g(g(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(g(b)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(f(a)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(b))))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(g(b)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1293) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes. ---------------------------------------- (1294) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(g(b)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1295) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP(up(f(b))) -> TOP(up(f(g(b)))) TOP(up(f(f(b)))) -> TOP(up(f(f(g(b))))) TOP(up(f(f(f(b))))) -> TOP(up(f(f(f(g(b)))))) TOP(up(f(f(f(f(b)))))) -> TOP(up(f(f(f(f(g(b))))))) TOP(up(f(f(f(f(f(b))))))) -> TOP(up(f(f(f(f(f(g(b)))))))) TOP(up(f(f(f(f(f(f(b)))))))) -> TOP(up(f(f(f(f(f(f(g(b))))))))) TOP(up(f(f(f(f(f(f(f(b))))))))) -> TOP(up(f(f(f(f(f(f(f(g(b)))))))))) TOP(up(f(f(f(f(f(f(f(f(b)))))))))) -> TOP(up(f(f(f(f(f(f(f(f(g(b))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(b))))))))))) -> TOP(up(f(f(f(f(f(f(f(f(f(g(b)))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(TOP(x_1)) = x_1 POL(a) = 0 POL(b) = 1 POL(c) = 0 POL(down(x_1)) = 0 POL(f(x_1)) = x_1 POL(f_flat(x_1)) = x_1 POL(fresh_constant) = 0 POL(g(x_1)) = 0 POL(g_flat(x_1)) = 0 POL(up(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x_1)) -> up(f(x_1)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) g_flat(up(x_1)) -> up(g(x_1)) ---------------------------------------- (1296) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(f(a)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1297) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP(up(g(a))) -> TOP(up(g(f(a)))) TOP(up(g(g(a)))) -> TOP(up(g(g(f(a))))) TOP(up(g(g(g(a))))) -> TOP(up(g(g(g(f(a)))))) TOP(up(g(g(g(g(a)))))) -> TOP(up(g(g(g(g(f(a))))))) TOP(up(g(g(g(g(g(a))))))) -> TOP(up(g(g(g(g(g(f(a)))))))) TOP(up(g(g(g(g(g(g(a)))))))) -> TOP(up(g(g(g(g(g(g(f(a))))))))) TOP(up(g(g(g(g(g(g(g(a))))))))) -> TOP(up(g(g(g(g(g(g(g(f(a)))))))))) TOP(up(g(g(g(g(g(g(g(g(a)))))))))) -> TOP(up(g(g(g(g(g(g(g(g(f(a))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(a))))))))))) -> TOP(up(g(g(g(g(g(g(g(g(g(f(a)))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(TOP(x_1)) = x_1 POL(a) = 1 POL(b) = 0 POL(c) = 0 POL(down(x_1)) = 0 POL(f(x_1)) = 0 POL(f_flat(x_1)) = 0 POL(fresh_constant) = 0 POL(g(x_1)) = x_1 POL(g_flat(x_1)) = x_1 POL(up(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: down(g(f(y6))) -> g_flat(down(f(y6))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) ---------------------------------------- (1298) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1299) MNOCProof (EQUIVALENT) We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set. ---------------------------------------- (1300) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (1301) SplitQDPProof (EQUIVALENT) We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem ---------------------------------------- (1302) Complex Obligation (AND) ---------------------------------------- (1303) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: down(a) -> up(f(a)) down(b) -> up(g(b)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(fresh_constant)) -> f_flat(down(fresh_constant)) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(fresh_constant)) -> g_flat(down(fresh_constant)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1304) SemLabProof (SOUND) We found the following model for the rules of the TRSs R and P. Interpretation over the domain with elements from 0 to 1. a: 0 b: 0 c: 0 down: 0 f: 0 fresh_constant: 1 up: 0 f_flat: 0 TOP: 0 g_flat: 0 g: 0 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. ---------------------------------------- (1305) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: down.0(a.) -> up.0(f.0(a.)) down.0(b.) -> up.0(g.0(b.)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x))))))))))) -> up.0(c.) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))))))) -> up.0(c.) down.0(f.0(a.)) -> f_flat.0(down.0(a.)) down.0(f.0(b.)) -> f_flat.0(down.0(b.)) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.)) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.)) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.1(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))))) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) The set Q consists of the following terms: down.0(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.1(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.0(g.1(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1306) 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: down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x))))))))))) -> up.0(c.) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))))))) -> up.0(c.) g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) Used ordering: POLO with Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(c.) = 0 POL(down.0(x_1)) = x_1 POL(down.1(x_1)) = 1 + x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 1 + x_1 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(g.0(x_1)) = x_1 POL(g.1(x_1)) = 1 + x_1 POL(g_flat.0(x_1)) = x_1 POL(up.0(x_1)) = x_1 POL(up.1(x_1)) = 1 + x_1 ---------------------------------------- (1307) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(a.)) -> f_flat.0(down.0(a.)) down.0(f.0(b.)) -> f_flat.0(down.0(b.)) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.1(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))))) down.0(b.) -> up.0(g.0(b.)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.1(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.0(g.1(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1308) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: down.0(f.1(fresh_constant.)) -> f_flat.0(down.1(fresh_constant.)) down.0(g.1(fresh_constant.)) -> g_flat.0(down.1(fresh_constant.)) Used ordering: Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(c.) = 0 POL(down.0(x_1)) = 1 + x_1 POL(down.1(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 1 + x_1 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(g.0(x_1)) = x_1 POL(g.1(x_1)) = x_1 POL(g_flat.0(x_1)) = x_1 POL(up.0(x_1)) = 1 + x_1 ---------------------------------------- (1309) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(a.)) -> f_flat.0(down.0(a.)) down.0(f.0(b.)) -> f_flat.0(down.0(b.)) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.1(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))))) down.0(b.) -> up.0(g.0(b.)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.1(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.0(g.1(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1310) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 18 less nodes. ---------------------------------------- (1311) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) The TRS R consists of the following rules: g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(a.)) -> f_flat.0(down.0(a.)) down.0(f.0(b.)) -> f_flat.0(down.0(b.)) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.1(fresh_constant.))) -> g_flat.0(down.0(g.1(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.1(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.1(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))))) down.0(b.) -> up.0(g.0(b.)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.1(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.0(g.1(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1312) PisEmptyProof (SOUND) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (1313) TRUE ---------------------------------------- (1314) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: g_flat(up(x_1)) -> up(g(x_1)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) f_flat(up(x_1)) -> up(f(x_1)) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) down(b) -> up(g(b)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1315) SplitQDPProof (EQUIVALENT) We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem ---------------------------------------- (1316) Complex Obligation (AND) ---------------------------------------- (1317) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: g_flat(up(x_1)) -> up(g(x_1)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(b)) -> f_flat(down(b)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) f_flat(up(x_1)) -> up(f(x_1)) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(b)) -> g_flat(down(b)) down(g(c)) -> g_flat(down(c)) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) down(b) -> up(g(b)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1318) SemLabProof (SOUND) We found the following model for the rules of the TRSs R and P. Interpretation over the domain with elements from 0 to 1. a: 0 b: 1 c: 0 down: 0 f: 0 fresh_constant: 0 up: 0 f_flat: 0 TOP: 0 g_flat: 0 g: 0 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. ---------------------------------------- (1319) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x))))))))))) -> up.0(c.) down.0(f.0(a.)) -> f_flat.0(down.0(a.)) down.0(f.1(b.)) -> f_flat.0(down.1(b.)) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.1(b.))) -> f_flat.0(down.0(f.1(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.1(b.)))) -> f_flat.0(down.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.1(b.))))) -> f_flat.0(down.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.1(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))))))) -> up.0(c.) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.1(b.)) -> g_flat.0(down.1(b.)) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.1(b.)))) -> g_flat.0(down.0(g.0(g.1(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.1(b.))))) -> g_flat.0(down.0(g.0(g.0(g.1(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) down.1(b.) -> up.0(g.1(b.)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.1(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.1(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.0(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.1(b.)) down.0(g.0(c.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.1(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.1(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.1(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.1(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.1(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1320) 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: g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x))))))))))) -> up.0(c.) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))))))) -> up.0(c.) Used ordering: POLO with Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(c.) = 0 POL(down.0(x_1)) = x_1 POL(down.1(x_1)) = 1 + x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 1 + x_1 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(g.0(x_1)) = x_1 POL(g.1(x_1)) = 1 + x_1 POL(g_flat.0(x_1)) = x_1 POL(up.0(x_1)) = x_1 POL(up.1(x_1)) = 1 + x_1 ---------------------------------------- (1321) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: down.0(f.1(b.)) -> f_flat.0(down.1(b.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.1(b.) -> up.0(g.1(b.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(a.)) -> f_flat.0(down.0(a.)) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.1(b.))) -> f_flat.0(down.0(f.1(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.1(b.)))) -> f_flat.0(down.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.1(b.))))) -> f_flat.0(down.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.1(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.1(b.)) -> g_flat.0(down.1(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.1(b.)))) -> g_flat.0(down.0(g.0(g.1(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.1(b.))))) -> g_flat.0(down.0(g.0(g.0(g.1(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.1(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.1(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.0(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.1(b.)) down.0(g.0(c.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.1(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.1(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.1(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.1(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.1(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1322) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: down.0(f.1(b.)) -> f_flat.0(down.1(b.)) Used ordering: Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(c.) = 0 POL(down.0(x_1)) = x_1 POL(down.1(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 1 + x_1 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(g.0(x_1)) = x_1 POL(g.1(x_1)) = x_1 POL(g_flat.0(x_1)) = x_1 POL(up.0(x_1)) = x_1 ---------------------------------------- (1323) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.1(b.) -> up.0(g.1(b.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(a.)) -> f_flat.0(down.0(a.)) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.1(b.))) -> f_flat.0(down.0(f.1(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.1(b.)))) -> f_flat.0(down.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.1(b.))))) -> f_flat.0(down.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.1(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.1(b.)) -> g_flat.0(down.1(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.1(b.)))) -> g_flat.0(down.0(g.0(g.1(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.1(b.))))) -> g_flat.0(down.0(g.0(g.0(g.1(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.1(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.1(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.0(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.1(b.)) down.0(g.0(c.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.1(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.1(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.1(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.1(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.1(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1324) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes. ---------------------------------------- (1325) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) The TRS R consists of the following rules: g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.1(b.) -> up.0(g.1(b.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(a.)) -> f_flat.0(down.0(a.)) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.1(b.))) -> f_flat.0(down.0(f.1(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.1(b.)))) -> f_flat.0(down.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.1(b.))))) -> f_flat.0(down.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.1(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.1(b.)) -> g_flat.0(down.1(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.1(b.)))) -> g_flat.0(down.0(g.0(g.1(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.1(b.))))) -> g_flat.0(down.0(g.0(g.0(g.1(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.1(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.1(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.0(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.1(b.)) down.0(g.0(c.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.1(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.1(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.1(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.1(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.1(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1326) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(c.) = 0 POL(down.0(x_1)) = 0 POL(down.1(x_1)) = 0 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 0 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(g.0(x_1)) = 0 POL(g.1(x_1)) = 1 POL(g_flat.0(x_1)) = 0 POL(up.0(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.1(b.)))) -> g_flat.0(down.0(g.0(g.1(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.1(b.))))) -> g_flat.0(down.0(g.0(g.0(g.1(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(g.1(b.)) -> g_flat.0(down.1(b.)) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) ---------------------------------------- (1327) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) The TRS R consists of the following rules: g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.1(b.) -> up.0(g.1(b.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(a.)) -> f_flat.0(down.0(a.)) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.1(b.))) -> f_flat.0(down.0(f.1(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.1(b.)))) -> f_flat.0(down.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.1(b.))))) -> f_flat.0(down.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.1(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.1(b.)) -> g_flat.0(down.1(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(a.)) -> g_flat.0(down.0(a.)) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.1(b.))) -> g_flat.0(down.0(g.1(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.1(b.)))) -> g_flat.0(down.0(g.0(g.1(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.1(b.))))) -> g_flat.0(down.0(g.0(g.0(g.1(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.1(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.1(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.0(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.1(b.)) down.0(g.0(c.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.1(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.1(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.1(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.1(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.1(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.1(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.1(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.1(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1328) PisEmptyProof (SOUND) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (1329) TRUE ---------------------------------------- (1330) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: g_flat(up(x_1)) -> up(g(x_1)) down(b) -> up(g(b)) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(b)) -> g_flat(down(b)) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(c)) -> g_flat(down(c)) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1331) SplitQDPProof (EQUIVALENT) We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem ---------------------------------------- (1332) Complex Obligation (AND) ---------------------------------------- (1333) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: g_flat(up(x_1)) -> up(g(x_1)) down(b) -> up(g(b)) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(f(a)) -> f_flat(down(a)) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(b)) -> g_flat(down(b)) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(g(a)) -> g_flat(down(a)) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(c)) -> g_flat(down(c)) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1334) SemLabProof (SOUND) We found the following model for the rules of the TRSs R and P. Interpretation over the domain with elements from 0 to 1. a: 1 b: 0 c: 0 down: 0 f: 0 fresh_constant: 0 up: 0 f_flat: 0 TOP: 0 g_flat: 0 g: 0 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. ---------------------------------------- (1335) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) down.0(b.) -> up.0(g.0(b.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x))))))))))) -> up.0(c.) down.0(f.1(a.)) -> f_flat.0(down.1(a.)) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.1(a.))) -> f_flat.0(down.0(f.1(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.1(a.)))) -> f_flat.0(down.0(f.0(f.1(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.1(a.))))) -> f_flat.0(down.0(f.0(f.0(f.1(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.1(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))))))) -> up.0(c.) down.0(g.1(a.)) -> g_flat.0(down.1(a.)) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.1(a.))) -> g_flat.0(down.0(g.1(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.1(a.)))) -> g_flat.0(down.0(g.0(g.1(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.1(a.))))) -> g_flat.0(down.0(g.0(g.0(g.1(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.1(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) down.1(a.) -> up.0(f.1(a.)) The set Q consists of the following terms: down.1(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.1(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.0(fresh_constant.)) down.0(g.1(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.1(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) down.0(g.0(g.1(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) down.0(f.0(f.0(f.1(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(g.0(g.0(g.1(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.1(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.1(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.1(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.1(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1336) 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: g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x))))))))))) -> up.0(c.) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))))))) -> up.0(c.) Used ordering: POLO with Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(c.) = 0 POL(down.0(x_1)) = x_1 POL(down.1(x_1)) = 1 + x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 1 + x_1 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(g.0(x_1)) = x_1 POL(g.1(x_1)) = 1 + x_1 POL(g_flat.0(x_1)) = x_1 POL(up.0(x_1)) = x_1 POL(up.1(x_1)) = 1 + x_1 ---------------------------------------- (1337) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: down.0(f.1(a.)) -> f_flat.0(down.1(a.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.1(a.) -> up.0(f.1(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.1(a.))) -> f_flat.0(down.0(f.1(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.1(a.)))) -> f_flat.0(down.0(f.0(f.1(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.1(a.))))) -> f_flat.0(down.0(f.0(f.0(f.1(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.1(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.1(a.)) -> g_flat.0(down.1(a.)) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.1(a.))) -> g_flat.0(down.0(g.1(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.1(a.)))) -> g_flat.0(down.0(g.0(g.1(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.1(a.))))) -> g_flat.0(down.0(g.0(g.0(g.1(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.1(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) down.0(b.) -> up.0(g.0(b.)) The set Q consists of the following terms: down.1(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.1(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.0(fresh_constant.)) down.0(g.1(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.1(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) down.0(g.0(g.1(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) down.0(f.0(f.0(f.1(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(g.0(g.0(g.1(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.1(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.1(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.1(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.1(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1338) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: down.0(g.1(a.)) -> g_flat.0(down.1(a.)) Used ordering: Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(c.) = 0 POL(down.0(x_1)) = 1 + x_1 POL(down.1(x_1)) = 1 + x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = x_1 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(g.0(x_1)) = x_1 POL(g.1(x_1)) = 1 + x_1 POL(g_flat.0(x_1)) = x_1 POL(up.0(x_1)) = 1 + x_1 ---------------------------------------- (1339) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: down.0(f.1(a.)) -> f_flat.0(down.1(a.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.1(a.) -> up.0(f.1(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.1(a.))) -> f_flat.0(down.0(f.1(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.1(a.)))) -> f_flat.0(down.0(f.0(f.1(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.1(a.))))) -> f_flat.0(down.0(f.0(f.0(f.1(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.1(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.1(a.))) -> g_flat.0(down.0(g.1(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.1(a.)))) -> g_flat.0(down.0(g.0(g.1(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.1(a.))))) -> g_flat.0(down.0(g.0(g.0(g.1(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.1(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) down.0(b.) -> up.0(g.0(b.)) The set Q consists of the following terms: down.1(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.1(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.0(fresh_constant.)) down.0(g.1(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.1(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) down.0(g.0(g.1(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) down.0(f.0(f.0(f.1(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(g.0(g.0(g.1(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.1(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.1(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.1(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.1(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1340) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes. ---------------------------------------- (1341) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: down.0(f.1(a.)) -> f_flat.0(down.1(a.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.1(a.) -> up.0(f.1(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.1(a.))) -> f_flat.0(down.0(f.1(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.1(a.)))) -> f_flat.0(down.0(f.0(f.1(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.1(a.))))) -> f_flat.0(down.0(f.0(f.0(f.1(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.1(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.1(a.))) -> g_flat.0(down.0(g.1(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.1(a.)))) -> g_flat.0(down.0(g.0(g.1(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.1(a.))))) -> g_flat.0(down.0(g.0(g.0(g.1(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.1(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) down.0(b.) -> up.0(g.0(b.)) The set Q consists of the following terms: down.1(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.1(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.0(fresh_constant.)) down.0(g.1(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.1(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) down.0(g.0(g.1(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) down.0(f.0(f.0(f.1(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(g.0(g.0(g.1(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.1(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.1(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.1(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.1(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1342) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(c.) = 0 POL(down.0(x_1)) = 0 POL(down.1(x_1)) = x_1 POL(f.0(x_1)) = 0 POL(f.1(x_1)) = 1 POL(f_flat.0(x_1)) = 0 POL(fresh_constant.) = 0 POL(g.0(x_1)) = x_1 POL(g.1(x_1)) = 0 POL(g_flat.0(x_1)) = x_1 POL(up.0(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.1(a.))) -> f_flat.0(down.0(f.1(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.1(a.)))) -> f_flat.0(down.0(f.0(f.1(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.1(a.))))) -> f_flat.0(down.0(f.0(f.0(f.1(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.1(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.0(f.1(a.)) -> f_flat.0(down.1(a.)) ---------------------------------------- (1343) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) The TRS R consists of the following rules: down.0(f.1(a.)) -> f_flat.0(down.1(a.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) down.1(a.) -> up.0(f.1(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(c.)) -> f_flat.0(down.0(c.)) down.0(f.0(f.1(a.))) -> f_flat.0(down.0(f.1(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.1(a.)))) -> f_flat.0(down.0(f.0(f.1(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.1(a.))))) -> f_flat.0(down.0(f.0(f.0(f.1(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.1(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(c.)) -> g_flat.0(down.0(c.)) down.0(g.0(g.1(a.))) -> g_flat.0(down.0(g.1(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.1(a.)))) -> g_flat.0(down.0(g.0(g.1(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.1(a.))))) -> g_flat.0(down.0(g.0(g.0(g.1(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.1(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) down.0(b.) -> up.0(g.0(b.)) The set Q consists of the following terms: down.1(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.1(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.0(c.)) down.0(f.0(fresh_constant.)) down.0(g.1(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.1(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.0(fresh_constant.))) down.0(g.0(g.1(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.0(g.0(g.0(fresh_constant.))) down.0(f.0(f.0(f.1(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(g.0(g.0(g.1(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.1(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.1(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.1(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.1(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1344) PisEmptyProof (SOUND) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (1345) TRUE ---------------------------------------- (1346) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: down(f(a)) -> f_flat(down(a)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(b)) -> g_flat(down(b)) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(c)) -> g_flat(down(c)) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) down(b) -> up(g(b)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1347) SplitQDPProof (EQUIVALENT) We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem ---------------------------------------- (1348) Complex Obligation (AND) ---------------------------------------- (1349) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: down(f(a)) -> f_flat(down(a)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(c)) -> f_flat(down(c)) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(b)) -> g_flat(down(b)) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(c)) -> g_flat(down(c)) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) down(b) -> up(g(b)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1350) SemLabProof (SOUND) We found the following model for the rules of the TRSs R and P. Interpretation over the domain with elements from 0 to 1. a: 0 b: 0 c: 1 down: 0 f: 0 fresh_constant: 0 up: 0 f_flat: 0 TOP: 0 g_flat: 0 g: 0 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. ---------------------------------------- (1351) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: down.0(f.0(a.)) -> f_flat.0(down.0(a.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) down.0(a.) -> up.0(f.0(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.1(c.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x))))))))))) -> up.1(c.) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.1(c.)) -> f_flat.0(down.1(c.)) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.1(c.)))) -> f_flat.0(down.0(f.0(f.1(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.1(c.))))) -> f_flat.0(down.0(f.0(f.0(f.1(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.1(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.1(c.) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))))))) -> up.1(c.) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.1(c.)) -> g_flat.0(down.1(c.)) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.1(c.))) -> g_flat.0(down.0(g.1(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.1(c.)))) -> g_flat.0(down.0(g.0(g.1(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.1(c.))))) -> g_flat.0(down.0(g.0(g.0(g.1(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.1(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) down.0(b.) -> up.0(g.0(b.)) The set Q consists of the following terms: down.0(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.1(c.)) down.0(f.0(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.1(c.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.1(c.))) down.0(f.0(f.0(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.1(c.))) down.0(g.0(g.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.1(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.1(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.1(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.1(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.1(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.1(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1352) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: down.0(f.1(c.)) -> f_flat.0(down.1(c.)) down.0(g.1(c.)) -> g_flat.0(down.1(c.)) Used ordering: Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(c.) = 0 POL(down.0(x_1)) = 1 + x_1 POL(down.1(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = x_1 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(g.0(x_1)) = x_1 POL(g.1(x_1)) = x_1 POL(g_flat.0(x_1)) = x_1 POL(up.0(x_1)) = 1 + x_1 POL(up.1(x_1)) = 1 + x_1 ---------------------------------------- (1353) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(f.0(g.1(x0)))) -> TOP.0(f_flat.0(down.0(g.1(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.1(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: down.0(f.0(a.)) -> f_flat.0(down.0(a.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) down.0(a.) -> up.0(f.0(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.1(c.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x))))))))))) -> up.1(c.) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.1(c.)))) -> f_flat.0(down.0(f.0(f.1(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.1(c.))))) -> f_flat.0(down.0(f.0(f.0(f.1(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.1(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.1(c.) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))))))) -> up.1(c.) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.1(c.))) -> g_flat.0(down.0(g.1(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.1(c.)))) -> g_flat.0(down.0(g.0(g.1(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.1(c.))))) -> g_flat.0(down.0(g.0(g.0(g.1(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.1(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) down.0(b.) -> up.0(g.0(b.)) The set Q consists of the following terms: down.0(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.1(c.)) down.0(f.0(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.1(c.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.1(c.))) down.0(f.0(f.0(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.1(c.))) down.0(g.0(g.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.1(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.1(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.1(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.1(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.1(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.1(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1354) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 18 less nodes. ---------------------------------------- (1355) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) The TRS R consists of the following rules: down.0(f.0(a.)) -> f_flat.0(down.0(a.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) g_flat.0(up.1(x_1)) -> up.0(g.1(x_1)) down.0(a.) -> up.0(f.0(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.1(c.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x))))))))))) -> up.1(c.) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(g.1(y4))) -> f_flat.0(down.0(g.1(y4))) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.0(g.1(y10)))) -> f_flat.0(down.0(f.0(g.1(y10)))) down.0(f.0(f.1(c.))) -> f_flat.0(down.0(f.1(c.))) down.0(f.0(f.0(fresh_constant.))) -> f_flat.0(down.0(f.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.0(g.1(y16))))) -> f_flat.0(down.0(f.0(f.0(g.1(y16))))) down.0(f.0(f.0(f.1(c.)))) -> f_flat.0(down.0(f.0(f.1(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.0(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.1(y22)))))) down.0(f.0(f.0(f.0(f.1(c.))))) -> f_flat.0(down.0(f.0(f.0(f.1(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.1(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.1(c.) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x))))))))))) -> up.1(c.) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.1(c.))) -> g_flat.0(down.0(g.1(c.))) down.0(g.0(g.0(fresh_constant.))) -> g_flat.0(down.0(g.0(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.1(c.)))) -> g_flat.0(down.0(g.0(g.1(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) -> g_flat.0(down.0(g.0(g.0(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.1(c.))))) -> g_flat.0(down.0(g.0(g.0(g.1(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) -> g_flat.0(down.0(g.0(g.0(g.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.1(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.1(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.1(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) down.0(b.) -> up.0(g.0(b.)) The set Q consists of the following terms: down.0(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.0(g.1(x0))) down.0(f.1(c.)) down.0(f.0(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.1(c.)) down.0(g.0(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.0(g.1(x0)))) down.0(f.0(f.1(c.))) down.0(f.0(f.0(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.1(c.))) down.0(g.0(g.0(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.0(g.1(x0))))) down.0(f.0(f.0(f.1(c.)))) down.0(f.0(f.0(f.0(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.1(c.)))) down.0(g.0(g.0(g.0(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.0(g.1(x0)))))) down.0(f.0(f.0(f.0(f.1(c.))))) down.0(f.0(f.0(f.0(f.0(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.1(c.))))) down.0(g.0(g.0(g.0(g.0(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.1(c.)))))) down.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.1(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.1(c.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1356) PisEmptyProof (SOUND) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (1357) TRUE ---------------------------------------- (1358) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: down(f(a)) -> f_flat(down(a)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(b)) -> g_flat(down(b)) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) down(b) -> up(g(b)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1359) SplitQDPProof (EQUIVALENT) We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem ---------------------------------------- (1360) Complex Obligation (AND) ---------------------------------------- (1361) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: down(f(a)) -> f_flat(down(a)) g_flat(up(x_1)) -> up(g(x_1)) down(a) -> up(f(a)) f_flat(up(x_1)) -> up(f(x_1)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(b)) -> g_flat(down(b)) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(fresh_constant))) -> g_flat(down(g(fresh_constant))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(fresh_constant)))) -> g_flat(down(g(g(fresh_constant)))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) -> g_flat(down(g(g(g(fresh_constant))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) -> g_flat(down(g(g(g(g(fresh_constant)))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) -> g_flat(down(g(g(g(g(g(fresh_constant))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) -> g_flat(down(g(g(g(g(g(g(fresh_constant)))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) -> g_flat(down(g(g(g(g(g(g(g(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(fresh_constant)))))))))) down(b) -> up(g(b)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1362) SemLabProof (SOUND) We found the following model for the rules of the TRSs R and P. Interpretation over the domain with elements from 0 to 1. a: 0 b: 0 c: 0 down: 0 f: 0 fresh_constant: 1 up: 0 f_flat: 0 TOP: 0 g_flat: 0 g: x0 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. ---------------------------------------- (1363) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(f.1(g.1(x0)))) -> TOP.0(f_flat.0(down.1(g.1(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.1(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.1(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(f.0(f.0(f.1(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.1(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: down.0(f.0(a.)) -> f_flat.0(down.0(a.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) g_flat.0(up.1(x_1)) -> up.1(g.1(x_1)) down.0(a.) -> up.0(f.0(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x))))))))))) -> up.0(c.) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.1(g.1(y4))) -> f_flat.0(down.1(g.1(y4))) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.1(g.1(y10)))) -> f_flat.0(down.0(f.1(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.1(g.1(y16))))) -> f_flat.0(down.0(f.0(f.1(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.1(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.1(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.1(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(x))))))))))) -> up.0(c.) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.1(g.1(g.1(g.1(fresh_constant.)))) -> g_flat.0(down.1(g.1(g.1(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.1(g.1(g.1(g.1(g.1(fresh_constant.))))) -> g_flat.0(down.1(g.1(g.1(g.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))))) down.0(b.) -> up.0(g.0(b.)) The set Q consists of the following terms: down.0(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.1(g.1(x0))) down.0(f.0(c.)) down.0(f.1(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.1(g.1(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.1(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.1(g.1(g.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.1(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.1(g.1(g.1(g.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.1(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.1(g.1(g.1(g.1(g.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1364) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: TOP.0(up.0(f.1(g.1(x0)))) -> TOP.0(f_flat.0(down.1(g.1(x0)))) TOP.0(up.0(f.0(f.1(g.1(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.1(g.1(x0))))) TOP.0(up.0(f.0(f.0(f.1(g.1(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.1(g.1(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.1(g.1(x0)))))))))))) Strictly oriented rules of the TRS R: down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x))))))))))) -> up.0(c.) down.0(f.1(g.1(y4))) -> f_flat.0(down.1(g.1(y4))) Used ordering: Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(c.) = 0 POL(down.0(x_1)) = x_1 POL(down.1(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = 1 + x_1 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(g.0(x_1)) = x_1 POL(g.1(x_1)) = x_1 POL(g_flat.0(x_1)) = x_1 POL(up.0(x_1)) = x_1 POL(up.1(x_1)) = 1 + x_1 ---------------------------------------- (1365) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(f.1(x0)))) -> TOP.0(g_flat.0(down.0(f.1(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(g.0(g.0(f.1(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.1(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.1(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.1(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.1(x0)))))))))))) The TRS R consists of the following rules: down.0(f.0(a.)) -> f_flat.0(down.0(a.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) g_flat.0(up.1(x_1)) -> up.1(g.1(x_1)) down.0(a.) -> up.0(f.0(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.1(g.1(y10)))) -> f_flat.0(down.0(f.1(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.1(g.1(y16))))) -> f_flat.0(down.0(f.0(f.1(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.1(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.1(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.1(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(x))))))))))) -> up.0(c.) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.1(g.1(g.1(g.1(fresh_constant.)))) -> g_flat.0(down.1(g.1(g.1(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.1(g.1(g.1(g.1(g.1(fresh_constant.))))) -> g_flat.0(down.1(g.1(g.1(g.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))))) down.0(b.) -> up.0(g.0(b.)) The set Q consists of the following terms: down.0(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.1(g.1(x0))) down.0(f.0(c.)) down.0(f.1(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.1(g.1(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.1(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.1(g.1(g.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.1(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.1(g.1(g.1(g.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.1(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.1(g.1(g.1(g.1(g.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1366) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes. ---------------------------------------- (1367) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) The TRS R consists of the following rules: down.0(f.0(a.)) -> f_flat.0(down.0(a.)) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) g_flat.0(up.1(x_1)) -> up.1(g.1(x_1)) down.0(a.) -> up.0(f.0(a.)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.1(g.1(y10)))) -> f_flat.0(down.0(f.1(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.1(g.1(y16))))) -> f_flat.0(down.0(f.0(f.1(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.1(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.1(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.1(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(x))))))))))) -> up.0(c.) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.1(g.1(g.1(g.1(fresh_constant.)))) -> g_flat.0(down.1(g.1(g.1(fresh_constant.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.1(g.1(g.1(g.1(g.1(fresh_constant.))))) -> g_flat.0(down.1(g.1(g.1(g.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))))) down.0(b.) -> up.0(g.0(b.)) The set Q consists of the following terms: down.0(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.1(g.1(x0))) down.0(f.0(c.)) down.0(f.1(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.1(g.1(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.1(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.1(g.1(g.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.1(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.1(g.1(g.1(g.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.1(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.1(g.1(g.1(g.1(g.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1368) 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: down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(x))))))))))) -> up.0(c.) down.1(g.1(g.1(fresh_constant.))) -> g_flat.0(down.1(g.1(fresh_constant.))) down.1(g.1(g.1(g.1(fresh_constant.)))) -> g_flat.0(down.1(g.1(g.1(fresh_constant.)))) down.1(g.1(g.1(g.1(g.1(fresh_constant.))))) -> g_flat.0(down.1(g.1(g.1(g.1(fresh_constant.))))) down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))))) -> g_flat.0(down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))))) Used ordering: POLO with Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(c.) = 0 POL(down.0(x_1)) = 1 + x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = x_1 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(g.0(x_1)) = x_1 POL(g.1(x_1)) = x_1 POL(g_flat.0(x_1)) = x_1 POL(up.0(x_1)) = 1 + x_1 POL(up.1(x_1)) = 1 + x_1 ---------------------------------------- (1369) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) The TRS R consists of the following rules: down.0(f.0(a.)) -> f_flat.0(down.0(a.)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.1(g.1(y10)))) -> f_flat.0(down.0(f.1(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.1(g.1(y16))))) -> f_flat.0(down.0(f.0(f.1(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.1(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.1(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.1(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) g_flat.0(up.1(x_1)) -> up.1(g.1(x_1)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(b.) -> up.0(g.0(b.)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.1(g.1(x0))) down.0(f.0(c.)) down.0(f.1(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.1(g.1(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.1(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.1(g.1(g.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.1(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.1(g.1(g.1(g.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.1(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.1(g.1(g.1(g.1(g.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1370) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented rules of the TRS R: f_flat.0(up.1(x_1)) -> up.0(f.1(x_1)) Used ordering: Polynomial interpretation [POLO]: POL(TOP.0(x_1)) = x_1 POL(a.) = 0 POL(b.) = 0 POL(c.) = 0 POL(down.0(x_1)) = x_1 POL(f.0(x_1)) = x_1 POL(f.1(x_1)) = x_1 POL(f_flat.0(x_1)) = x_1 POL(fresh_constant.) = 0 POL(g.0(x_1)) = x_1 POL(g.1(x_1)) = x_1 POL(g_flat.0(x_1)) = x_1 POL(up.0(x_1)) = x_1 POL(up.1(x_1)) = 1 + x_1 ---------------------------------------- (1371) Obligation: Q DP problem: The TRS P consists of the following rules: TOP.0(up.0(f.0(g.0(x0)))) -> TOP.0(f_flat.0(down.0(g.0(x0)))) TOP.0(up.0(g.0(f.0(x0)))) -> TOP.0(g_flat.0(down.0(f.0(x0)))) TOP.0(up.0(g.0(g.0(f.0(x0))))) -> TOP.0(g_flat.0(g_flat.0(down.0(f.0(x0))))) TOP.0(up.0(f.0(f.0(g.0(x0))))) -> TOP.0(f_flat.0(f_flat.0(down.0(g.0(x0))))) TOP.0(up.0(f.0(f.0(f.0(g.0(x0)))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))) TOP.0(up.0(g.0(g.0(g.0(f.0(x0)))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))) TOP.0(up.0(f.0(f.0(f.0(f.0(g.0(x0))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(f.0(x0))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0))))))))))) TOP.0(up.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))))) -> TOP.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(f_flat.0(down.0(g.0(x0)))))))))))) TOP.0(up.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))))) -> TOP.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(g_flat.0(down.0(f.0(x0)))))))))))) The TRS R consists of the following rules: down.0(f.0(a.)) -> f_flat.0(down.0(a.)) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x))))))))))) -> up.0(c.) down.0(f.0(g.0(y4))) -> f_flat.0(down.0(g.0(y4))) down.0(f.0(f.0(a.))) -> f_flat.0(down.0(f.0(a.))) down.0(f.0(f.0(b.))) -> f_flat.0(down.0(f.0(b.))) down.0(f.0(f.0(g.0(y10)))) -> f_flat.0(down.0(f.0(g.0(y10)))) down.0(f.0(f.1(g.1(y10)))) -> f_flat.0(down.0(f.1(g.1(y10)))) down.0(f.0(f.0(c.))) -> f_flat.0(down.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) -> f_flat.0(down.0(f.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) -> f_flat.0(down.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) -> f_flat.0(down.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(y16))))) -> f_flat.0(down.0(f.0(f.0(g.0(y16))))) down.0(f.0(f.0(f.1(g.1(y16))))) -> f_flat.0(down.0(f.0(f.1(g.1(y16))))) down.0(f.0(f.0(f.0(c.)))) -> f_flat.0(down.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) -> f_flat.0(down.0(f.0(f.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) -> f_flat.0(down.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) -> f_flat.0(down.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.0(g.0(y22)))))) down.0(f.0(f.0(f.0(f.1(g.1(y22)))))) -> f_flat.0(down.0(f.0(f.0(f.1(g.1(y22)))))) down.0(f.0(f.0(f.0(f.0(c.))))) -> f_flat.0(down.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) -> f_flat.0(down.0(f.0(f.0(f.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(g.0(y28))))))) down.0(f.0(f.0(f.0(f.0(f.1(g.1(y28))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(g.1(y28))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(g.0(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y34)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(g.1(y34)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y40))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y40))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y46)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y46)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y52))))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(y52))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) -> f_flat.0(down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) g_flat.0(up.0(x_1)) -> up.0(g.0(x_1)) g_flat.0(up.1(x_1)) -> up.1(g.1(x_1)) f_flat.0(up.0(x_1)) -> up.0(f.0(x_1)) down.0(g.0(b.)) -> g_flat.0(down.0(b.)) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x))))))))))) -> up.0(c.) down.0(g.0(f.0(y6))) -> g_flat.0(down.0(f.0(y6))) down.0(g.0(f.1(y6))) -> g_flat.0(down.0(f.1(y6))) down.0(g.0(g.0(a.))) -> g_flat.0(down.0(g.0(a.))) down.0(g.0(g.0(f.0(y12)))) -> g_flat.0(down.0(g.0(f.0(y12)))) down.0(g.0(g.0(f.1(y12)))) -> g_flat.0(down.0(g.0(f.1(y12)))) down.0(g.0(g.0(b.))) -> g_flat.0(down.0(g.0(b.))) down.0(g.0(g.0(c.))) -> g_flat.0(down.0(g.0(c.))) down.0(g.0(g.0(g.0(a.)))) -> g_flat.0(down.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(y18))))) -> g_flat.0(down.0(g.0(g.0(f.0(y18))))) down.0(g.0(g.0(g.0(f.1(y18))))) -> g_flat.0(down.0(g.0(g.0(f.1(y18))))) down.0(g.0(g.0(g.0(b.)))) -> g_flat.0(down.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) -> g_flat.0(down.0(g.0(g.0(c.)))) down.0(g.0(g.0(g.0(g.0(a.))))) -> g_flat.0(down.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.0(y24)))))) down.0(g.0(g.0(g.0(g.0(f.1(y24)))))) -> g_flat.0(down.0(g.0(g.0(g.0(f.1(y24)))))) down.0(g.0(g.0(g.0(g.0(b.))))) -> g_flat.0(down.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) -> g_flat.0(down.0(g.0(g.0(g.0(c.))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.0(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(y30))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(f.1(y30))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(c.)))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.0(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(f.1(y36)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y42))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y48)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(y54))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) -> g_flat.0(down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.0(b.) -> up.0(g.0(b.)) down.0(a.) -> up.0(f.0(a.)) The set Q consists of the following terms: down.0(a.) down.0(b.) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(x0))))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(x0))))))))))) down.0(f.0(a.)) down.0(f.0(b.)) down.0(f.0(g.0(x0))) down.0(f.1(g.1(x0))) down.0(f.0(c.)) down.0(f.1(fresh_constant.)) down.0(g.0(a.)) down.0(g.0(f.0(x0))) down.0(g.0(f.1(x0))) down.0(g.0(b.)) down.0(g.0(c.)) down.1(g.1(fresh_constant.)) down.0(f.0(f.0(a.))) down.0(f.0(f.0(b.))) down.0(f.0(f.0(g.0(x0)))) down.0(f.0(f.1(g.1(x0)))) down.0(f.0(f.0(c.))) down.0(f.0(f.1(fresh_constant.))) down.0(g.0(g.0(a.))) down.0(g.0(g.0(f.0(x0)))) down.0(g.0(g.0(f.1(x0)))) down.0(g.0(g.0(b.))) down.0(g.0(g.0(c.))) down.1(g.1(g.1(fresh_constant.))) down.0(f.0(f.0(f.0(a.)))) down.0(f.0(f.0(f.0(b.)))) down.0(f.0(f.0(f.0(g.0(x0))))) down.0(f.0(f.0(f.1(g.1(x0))))) down.0(f.0(f.0(f.0(c.)))) down.0(f.0(f.0(f.1(fresh_constant.)))) down.0(g.0(g.0(g.0(a.)))) down.0(g.0(g.0(g.0(f.0(x0))))) down.0(g.0(g.0(g.0(f.1(x0))))) down.0(g.0(g.0(g.0(b.)))) down.0(g.0(g.0(g.0(c.)))) down.1(g.1(g.1(g.1(fresh_constant.)))) down.0(f.0(f.0(f.0(f.0(a.))))) down.0(f.0(f.0(f.0(f.0(b.))))) down.0(f.0(f.0(f.0(f.0(g.0(x0)))))) down.0(f.0(f.0(f.0(f.1(g.1(x0)))))) down.0(f.0(f.0(f.0(f.0(c.))))) down.0(f.0(f.0(f.0(f.1(fresh_constant.))))) down.0(g.0(g.0(g.0(g.0(a.))))) down.0(g.0(g.0(g.0(g.0(f.0(x0)))))) down.0(g.0(g.0(g.0(g.0(f.1(x0)))))) down.0(g.0(g.0(g.0(g.0(b.))))) down.0(g.0(g.0(g.0(g.0(c.))))) down.1(g.1(g.1(g.1(g.1(fresh_constant.))))) down.0(f.0(f.0(f.0(f.0(f.0(a.)))))) down.0(f.0(f.0(f.0(f.0(f.0(b.)))))) down.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))) down.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))) down.0(f.0(f.0(f.0(f.0(f.0(c.)))))) down.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))) down.0(g.0(g.0(g.0(g.0(g.0(a.)))))) down.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))) down.0(g.0(g.0(g.0(g.0(g.0(b.)))))) down.0(g.0(g.0(g.0(g.0(g.0(c.)))))) down.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(a.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(b.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(g.0(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(g.1(x0))))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(c.)))))))))) down.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.0(f.1(fresh_constant.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(a.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.0(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(f.1(x0))))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(b.)))))))))) down.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(g.0(c.)))))))))) down.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(g.1(fresh_constant.)))))))))) f_flat.0(up.0(x0)) f_flat.0(up.1(x0)) g_flat.0(up.0(x0)) g_flat.0(up.1(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1372) PisEmptyProof (SOUND) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (1373) TRUE ---------------------------------------- (1374) Obligation: Q DP problem: The TRS P consists of the following rules: TOP(up(f(g(x0)))) -> TOP(f_flat(down(g(x0)))) TOP(up(g(f(x0)))) -> TOP(g_flat(down(f(x0)))) TOP(up(g(g(f(x0))))) -> TOP(g_flat(g_flat(down(f(x0))))) TOP(up(f(f(g(x0))))) -> TOP(f_flat(f_flat(down(g(x0))))) TOP(up(f(f(f(g(x0)))))) -> TOP(f_flat(f_flat(f_flat(down(g(x0)))))) TOP(up(g(g(g(f(x0)))))) -> TOP(g_flat(g_flat(g_flat(down(f(x0)))))) TOP(up(f(f(f(f(g(x0))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))) TOP(up(g(g(g(g(f(x0))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))) TOP(up(f(f(f(f(f(g(x0)))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))) TOP(up(g(g(g(g(g(f(x0)))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))) TOP(up(f(f(f(f(f(f(g(x0))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))) TOP(up(g(g(g(g(g(g(f(x0))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))) TOP(up(f(f(f(f(f(f(f(g(x0)))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))) TOP(up(g(g(g(g(g(g(g(f(x0)))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))) TOP(up(f(f(f(f(f(f(f(f(g(x0))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0))))))))))) TOP(up(g(g(g(g(g(g(g(g(f(x0))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0))))))))))) TOP(up(f(f(f(f(f(f(f(f(f(g(x0)))))))))))) -> TOP(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(f_flat(down(g(x0)))))))))))) TOP(up(g(g(g(g(g(g(g(g(g(f(x0)))))))))))) -> TOP(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(g_flat(down(f(x0)))))))))))) The TRS R consists of the following rules: down(f(a)) -> f_flat(down(a)) down(f(f(f(f(f(f(f(f(f(f(x))))))))))) -> up(c) down(f(g(y4))) -> f_flat(down(g(y4))) down(f(f(a))) -> f_flat(down(f(a))) down(f(f(b))) -> f_flat(down(f(b))) down(f(f(g(y10)))) -> f_flat(down(f(g(y10)))) down(f(f(c))) -> f_flat(down(f(c))) down(f(f(fresh_constant))) -> f_flat(down(f(fresh_constant))) down(f(f(f(a)))) -> f_flat(down(f(f(a)))) down(f(f(f(b)))) -> f_flat(down(f(f(b)))) down(f(f(f(g(y16))))) -> f_flat(down(f(f(g(y16))))) down(f(f(f(c)))) -> f_flat(down(f(f(c)))) down(f(f(f(fresh_constant)))) -> f_flat(down(f(f(fresh_constant)))) down(f(f(f(f(a))))) -> f_flat(down(f(f(f(a))))) down(f(f(f(f(b))))) -> f_flat(down(f(f(f(b))))) down(f(f(f(f(g(y22)))))) -> f_flat(down(f(f(f(g(y22)))))) down(f(f(f(f(c))))) -> f_flat(down(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) -> f_flat(down(f(f(f(fresh_constant))))) down(f(f(f(f(f(a)))))) -> f_flat(down(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) -> f_flat(down(f(f(f(f(b)))))) down(f(f(f(f(f(g(y28))))))) -> f_flat(down(f(f(f(f(g(y28))))))) down(f(f(f(f(f(c)))))) -> f_flat(down(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) -> f_flat(down(f(f(f(f(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) -> f_flat(down(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) -> f_flat(down(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(y34)))))))) -> f_flat(down(f(f(f(f(f(g(y34)))))))) down(f(f(f(f(f(f(c))))))) -> f_flat(down(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) -> f_flat(down(f(f(f(f(f(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) -> f_flat(down(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) -> f_flat(down(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(y40))))))))) -> f_flat(down(f(f(f(f(f(f(g(y40))))))))) down(f(f(f(f(f(f(f(c)))))))) -> f_flat(down(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) -> f_flat(down(f(f(f(f(f(f(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) -> f_flat(down(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) -> f_flat(down(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(y46)))))))))) -> f_flat(down(f(f(f(f(f(f(f(g(y46)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) -> f_flat(down(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) -> f_flat(down(f(f(f(f(f(f(f(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(y52))))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(g(y52))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) -> f_flat(down(f(f(f(f(f(f(f(f(fresh_constant)))))))))) g_flat(up(x_1)) -> up(g(x_1)) f_flat(up(x_1)) -> up(f(x_1)) down(g(b)) -> g_flat(down(b)) down(g(g(g(g(g(g(g(g(g(g(x))))))))))) -> up(c) down(g(f(y6))) -> g_flat(down(f(y6))) down(g(g(a))) -> g_flat(down(g(a))) down(g(g(f(y12)))) -> g_flat(down(g(f(y12)))) down(g(g(b))) -> g_flat(down(g(b))) down(g(g(c))) -> g_flat(down(g(c))) down(g(g(g(a)))) -> g_flat(down(g(g(a)))) down(g(g(g(f(y18))))) -> g_flat(down(g(g(f(y18))))) down(g(g(g(b)))) -> g_flat(down(g(g(b)))) down(g(g(g(c)))) -> g_flat(down(g(g(c)))) down(g(g(g(g(a))))) -> g_flat(down(g(g(g(a))))) down(g(g(g(g(f(y24)))))) -> g_flat(down(g(g(g(f(y24)))))) down(g(g(g(g(b))))) -> g_flat(down(g(g(g(b))))) down(g(g(g(g(c))))) -> g_flat(down(g(g(g(c))))) down(g(g(g(g(g(a)))))) -> g_flat(down(g(g(g(g(a)))))) down(g(g(g(g(g(f(y30))))))) -> g_flat(down(g(g(g(g(f(y30))))))) down(g(g(g(g(g(b)))))) -> g_flat(down(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) -> g_flat(down(g(g(g(g(c)))))) down(g(g(g(g(g(g(a))))))) -> g_flat(down(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(y36)))))))) -> g_flat(down(g(g(g(g(g(f(y36)))))))) down(g(g(g(g(g(g(b))))))) -> g_flat(down(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) -> g_flat(down(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(g(a)))))))) -> g_flat(down(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(y42))))))))) -> g_flat(down(g(g(g(g(g(g(f(y42))))))))) down(g(g(g(g(g(g(g(b)))))))) -> g_flat(down(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) -> g_flat(down(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(g(a))))))))) -> g_flat(down(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(y48)))))))))) -> g_flat(down(g(g(g(g(g(g(g(f(y48)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) -> g_flat(down(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) -> g_flat(down(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(y54))))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(f(y54))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) -> g_flat(down(g(g(g(g(g(g(g(g(c)))))))))) down(b) -> up(g(b)) down(a) -> up(f(a)) The set Q consists of the following terms: down(a) down(b) down(f(f(f(f(f(f(f(f(f(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(g(x0))))))))))) down(f(a)) down(f(b)) down(f(g(x0))) down(f(c)) down(f(fresh_constant)) down(g(a)) down(g(f(x0))) down(g(b)) down(g(c)) down(g(fresh_constant)) down(f(f(a))) down(f(f(b))) down(f(f(g(x0)))) down(f(f(c))) down(f(f(fresh_constant))) down(g(g(a))) down(g(g(f(x0)))) down(g(g(b))) down(g(g(c))) down(g(g(fresh_constant))) down(f(f(f(a)))) down(f(f(f(b)))) down(f(f(f(g(x0))))) down(f(f(f(c)))) down(f(f(f(fresh_constant)))) down(g(g(g(a)))) down(g(g(g(f(x0))))) down(g(g(g(b)))) down(g(g(g(c)))) down(g(g(g(fresh_constant)))) down(f(f(f(f(a))))) down(f(f(f(f(b))))) down(f(f(f(f(g(x0)))))) down(f(f(f(f(c))))) down(f(f(f(f(fresh_constant))))) down(g(g(g(g(a))))) down(g(g(g(g(f(x0)))))) down(g(g(g(g(b))))) down(g(g(g(g(c))))) down(g(g(g(g(fresh_constant))))) down(f(f(f(f(f(a)))))) down(f(f(f(f(f(b)))))) down(f(f(f(f(f(g(x0))))))) down(f(f(f(f(f(c)))))) down(f(f(f(f(f(fresh_constant)))))) down(g(g(g(g(g(a)))))) down(g(g(g(g(g(f(x0))))))) down(g(g(g(g(g(b)))))) down(g(g(g(g(g(c)))))) down(g(g(g(g(g(fresh_constant)))))) down(f(f(f(f(f(f(a))))))) down(f(f(f(f(f(f(b))))))) down(f(f(f(f(f(f(g(x0)))))))) down(f(f(f(f(f(f(c))))))) down(f(f(f(f(f(f(fresh_constant))))))) down(g(g(g(g(g(g(a))))))) down(g(g(g(g(g(g(f(x0)))))))) down(g(g(g(g(g(g(b))))))) down(g(g(g(g(g(g(c))))))) down(g(g(g(g(g(g(fresh_constant))))))) down(f(f(f(f(f(f(f(a)))))))) down(f(f(f(f(f(f(f(b)))))))) down(f(f(f(f(f(f(f(g(x0))))))))) down(f(f(f(f(f(f(f(c)))))))) down(f(f(f(f(f(f(f(fresh_constant)))))))) down(g(g(g(g(g(g(g(a)))))))) down(g(g(g(g(g(g(g(f(x0))))))))) down(g(g(g(g(g(g(g(b)))))))) down(g(g(g(g(g(g(g(c)))))))) down(g(g(g(g(g(g(g(fresh_constant)))))))) down(f(f(f(f(f(f(f(f(a))))))))) down(f(f(f(f(f(f(f(f(b))))))))) down(f(f(f(f(f(f(f(f(g(x0)))))))))) down(f(f(f(f(f(f(f(f(c))))))))) down(f(f(f(f(f(f(f(f(fresh_constant))))))))) down(g(g(g(g(g(g(g(g(a))))))))) down(g(g(g(g(g(g(g(g(f(x0)))))))))) down(g(g(g(g(g(g(g(g(b))))))))) down(g(g(g(g(g(g(g(g(c))))))))) down(g(g(g(g(g(g(g(g(fresh_constant))))))))) down(f(f(f(f(f(f(f(f(f(a)))))))))) down(f(f(f(f(f(f(f(f(f(b)))))))))) down(f(f(f(f(f(f(f(f(f(g(x0))))))))))) down(f(f(f(f(f(f(f(f(f(c)))))))))) down(f(f(f(f(f(f(f(f(f(fresh_constant)))))))))) down(g(g(g(g(g(g(g(g(g(a)))))))))) down(g(g(g(g(g(g(g(g(g(f(x0))))))))))) down(g(g(g(g(g(g(g(g(g(b)))))))))) down(g(g(g(g(g(g(g(g(g(c)))))))))) down(g(g(g(g(g(g(g(g(g(fresh_constant)))))))))) f_flat(up(x0)) g_flat(up(x0)) We have to consider all minimal (P,Q,R)-chains.